Math 489/889 Stochastic Processes and Advanced Mathematical Finance Homework 6 Steve Dunbar Due Monday, October 13, 2010
<Text-field style="Heading 1" layout="Heading 1">Problem 1</Text-field> PkkpRHVyYXRpb25HNiIsJiomSSNUMEdGJCIiIiwmSSJxR0YkRihJInBHRiQhIiJGLEYoKipJImFHRiRGKEYpRiwsJkYoRigpKiZGKkYoRitGLEYnRixGKCwmRihGKClGMUYuRixGLEYs LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqJkkjVDBHNiIiIiIsJkkicUdGJUYmSSJwR0YlISIiRipGJioqSSJhR0YlRiZGJ0YqLCZGJkYmKSomRihGJkYpRipGJEYqRiYsJkYmRiYpRi9GLEYqRipGKg== JSFH
<Text-field style="Heading 2" layout="Heading 2">Part a</Text-field> For \$ T_0 = 10 \$ and \$ a = 20 \$ , draw a graph of theduration as a function of the probability \$ q \$ . 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<Text-field style="Heading 2" layout="Heading 2">Part b</Text-field> For \$ a = 20 \$ and \$ q = 0.55 \$ draw a graph of the expected duration as a function of \$ T_0 \$ . 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<Text-field style="Heading 2" layout="Heading 2">Part c</Text-field> For \$ a = 20 \$ and \$ q = 0.45 \$ draw a graph of the expected duration as a function of \$ T_0 \$ . 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<Text-field style="Heading 1" layout="Heading 1">Problem 2</Text-field> The boundary condition at state 26 says that the duration of the "game" from 26 down to 18 is the sum of hte duration of the subsequent games from 26 down to 25, and then from 25 to 18. We can compute the duration of the first sub-game using Corollary 1. Then the set of 9 first-step equations in 9 unknowns is: QzQ+SSZlcW4yNkc2Ii9JJEQyNkdGJSwmKiQtSSIrRyUqcHJvdGVjdGVkRzYkIyIiIyIiJCMhIiJGMEYyIiIiSSREMjVHRiVGM0YzPkkmZXFuMjVHRiUvRjQsKEYnI0YzRjBJJEQyNEdGJUYuRjNGM0YzPkkmZXFuMjRHRiUvRjosKEY0RjlJJEQyM0dGJUYuRjNGM0YzPkkmZXFuMjNHRiUvRj8sKEY6RjlJJEQyMkdGJUYuRjNGM0YzPkkmZXFuMjJHRiUvRkQsKEY/RjlJJEQyMUdGJUYuRjNGM0YzPkkmZXFuMjFHRiUvRkksKEZEIyIiKiIjNUkkRDIwR0YlI0YzRlBGM0YzRjM+SSZlcW4yMEdGJS9GUSwoRkkjRjAiIiVJJEQxOUdGJSNGM0ZYRjNGM0YzPkkmZXFuMTlHRiUvRlksKEZRRldJJEQxOEdGJUZaRjNGM0YzPkkmZXFuMThHRiUvRmluIiIhRjM= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder L0kkRDI2RzYiLCYiIiQiIiJJJEQyNUdGJEYn L0kkRDI1RzYiLChJJEQyNkdGJCMiIiIiIiRJJEQyNEdGJCMiIiNGKUYoRig= L0kkRDI0RzYiLChJJEQyNUdGJCMiIiIiIiRJJEQyM0dGJCMiIiNGKUYoRig= L0kkRDIzRzYiLChJJEQyNEdGJCMiIiIiIiRJJEQyMkdGJCMiIiNGKUYoRig= L0kkRDIyRzYiLChJJEQyM0dGJCMiIiIiIiRJJEQyMUdGJCMiIiNGKUYoRig= L0kkRDIxRzYiLChJJEQyMkdGJCMiIioiIzVJJEQyMEdGJCMiIiJGKUYsRiw= L0kkRDIwRzYiLChJJEQyMUdGJCMiIiQiIiVJJEQxOUdGJCMiIiJGKUYsRiw= L0kkRDE5RzYiLChJJEQyMEdGJCMiIiQiIiVJJEQxOEdGJCMiIiJGKUYsRiw= L0kkRDE4RzYiIiIh QyQ+SSpkdXJhdGlvbnNHNiItSSZzb2x2ZUdGJTYkPCtJJmVxbjE4R0YlSSZlcW4xOUdGJUkmZXFuMjBHRiVJJmVxbjIxR0YlSSZlcW4yMkdGJUkmZXFuMjNHRiVJJmVxbjI0R0YlSSZlcW4yNUdGJUkmZXFuMjZHRiU8K0kkRDE4R0YlSSREMTlHRiVJJEQyMEdGJUkkRDIxR0YlSSREMjJHRiVJJEQyM0dGJUkkRDI0R0YlSSREMjVHRiVJJEQyNkdGJSIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder PCsvSSREMThHNiIiIiEvSSREMTlHRiUiJFwkL0kkRDIwR0YlIiRrJS9JJEQyMUdGJSIkLCYvSSREMjJHRiUiJC8mL0kkRDIzR0YlIiQyJi9JJEQyNEdGJSIkNSYvSSREMjVHRiUiJDgmL0kkRDI2R0YlIiQ7Jg== QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkSSIlRzYiSSREMjVHRigiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IiQ4Jg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Problem 3</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Insert the trial solution 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 into the difference equation. For s < kLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiLEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYtUSJ+RidGL0YyL0Y2RjRGOEY6RjxGPkZARkIvRkZGREYvthe substitution yields 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so the equation is satisfied by the trial particular solution. For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlMtRiw2JVEia0YnRi9GMkY5 the difference equation becomes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGNC1JI21vR0YkNi1RJyZzZG90O0YnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYkLUYjNiYtSSNtaUdGJDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0YnLUY4Ni1RKCZtaW51cztGJ0Y0RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5Gam4tRlQ2JVEic0YnRldGWkY0RjQtRjg2LVEifkYnRjRGO0Y+RkBGQkZERkZGSEZKRk0tRjg2LVEiPUYnRjRGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZmb0Zfby1GMTYkUSIwRidGNEZfby1GODYtUSIrRidGNEY7Rj5GQEZCRkRGRkZIRmluRltvRl9vLUYsNiQtRiM2JC1JJm1mcmFjR0YkNigtRjE2JFEiMUYnRjQtRiM2JUYwRldGWi8lLmxpbmV0aGlja25lc3NHRmdwLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl5xLyUpYmV2ZWxsZWRHRj1GNEY0RjdGK0Y3LUYsNiQtRiM2JkZTRmZuLUYsNiQtRiM2JkZcb0ZmbkZlcEY0RjRGNEY0Rl9vRltwRl9vRl5wRjdGK0Y3LUYsNiQtRiM2KUZTRl9vLUY4Ni1RKiZ1bWludXMwO0YnRjRGO0Y+RkBGQkZERkZGSEZpbkZbb0Zfby1GLDYkLUYjNiZGXG9GW3BGZXBGNEY0LUZUNiNRIUYnRjRGNEY0 and again the equation is satisfied identically. For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSJrRidGL0YyRjk=, the difference equation becomes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctRjM2LVEiPUYnRi9GNkY5RjtGPUY/RkFGQy9GRlEsMC4yNzc3Nzc4ZW1GJy9GSUZORjItRiw2JFEiMUYnRi9GMi1GMzYtUSIrRidGL0Y2RjlGO0Y9Rj9GQUZDL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRldGMi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbWZyYWNHRiQ2KEZQLUYjNiQtRiw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHRlIvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZG8vJSliZXZlbGxlZEdGOEYvRi8tRjM2LVEnJnNkb3Q7RidGL0Y2RjlGO0Y9Rj9GQUZDRkVGSC1GWjYkLUYjNiRGK0YvRi9GMkZTRjJGWUZpby1GWjYkRltvRi9GaW8tRlo2JC1GIzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GMzYtUSgmbWludXM7RidGL0Y2RjlGO0Y9Rj9GQUZDRlZGWC1GWjYkLUYjNiZGZnBGU0ZQRi9GL0YvRi8tRjM2LVEiLkYnRi9GNkY5RjtGPUY/RkFGQ0ZFRkhGLw== Therefore, the particular solution is 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= 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.
<Text-field style="Heading 1" layout="Heading 1">Problem 4</Text-field> PkkvRXhwZWN0ZWRWaXNpdHNHNiItSSIqRyUqcHJvdGVjdGVkRzYkIiIjNyMsJiooSSJzR0YkIiIiSSJTR0YkISIiLUkkc3VtR0YkNiQqJkkia0dGJEYuLCZGL0YuRjVGMEYuL0Y1O0YuLCZGL0YuRjBGLkYuRi4tRjI2JComRjVGLiwmRi1GLkY1RjBGLi9GNTtGLiwmRi1GLkYwRi5GMA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NyMsKCooSSJzRzYiIiIiSSJTRzYiISIiLCYqJEkiU0c2IiIiJCMiIiIiIidJIlNHNiIjISIiIiInIiIiIiIjKiRJInNHNiIiIiQjISIiIiIkSSJzRzYiIyIiIiIiJA== LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw== LUkkbWFwRyUqcHJvdGVjdGVkRzYlSSlzaW1wbGlmeUc2JEYkSShfc3lzbGliRzYiNyMsKCooSSJzR0YpIiIiSSJTR0YpISIiLCYqJEYvIiIkI0YuIiInRi8jRjBGNUYuIiIjKiRGLUYzI0YwRjNGLSNGLkYzLkkpc3ltYm9saWNHRik= NyMsJComSSJzRzYiIiIiLCYqJEkiU0c2IiIiIyIiIiokSSJzRzYiIiIjISIiIiIiIyIiIiIiJA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= PkksU3VtSW50ZWdlcnNHNiItSSRzdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JEkia0dGJC9GKzsiIiJJIk5HRiQ= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCgqJCwmSSJORzYiIiIiRidGJyIiIyNGJ0YoRiUjISIiRihGKkYn LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw== LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwoKiQsJkkiTkdGJyIiIkYtRi0iIiMjRi1GLkYsIyEiIkYuRjBGLS5JKXN5bWJvbGljR0Yn LCYqJEkiTkc2IiIiIyMiIiJGJkYkRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= PkkrU3VtU3F1YXJlc0c2Ii1JJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkKiRJImtHRiQiIiMvRiw7IiIiSSJOR0Yk LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); 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<Text-field style="Heading 1" layout="Heading 1">Problem 5</Text-field> PkkiQ0c2IioqLCZJIktHRiQiIiIqKkkickdGJEYoSSJTR0YkIiIkSSJ4R0YkRigsJkYoRigqJEYtIiIjISIiRigjRihGLEYoRishIiNGLUYxLCZGKEYoRi1GMUYx LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiosJkkiS0c2IiIiIioqSSJyR0YlRiZJIlNHRiUiIiRJInhHRiVGJiwmRiZGJiokRisiIiMhIiJGJiNGJkYqRiZGKSEiI0YrRi8sJkYmRiZGK0YvRi8= QyQtJkkmcGxvdHNHNiI2I0ksaW50ZXJhY3RpdmVHRiZGJiIiIg== JSFH
<Text-field style="Heading 2" layout="Heading 2">Part a</Text-field> QyQ+SSVkQ2R4RzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkiQ0dGJUkieEdGJSIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCgqKiwmKihJInJHNiIiIiJJIlNHRiciIiQsJkYoRigqJEkieEdGJyIiIyEiIkYoI0YoRioqKEYmRihGKUYqRi1GLiMhIiNGKkYoRilGM0YtRi8sJkYoRihGLUYvRi9GKCoqLCZJIktHRidGKCoqRiZGKEYpRipGLUYoRitGKEYwRihGKUYzRi1GM0Y0Ri9GLyoqRjZGKEYpRjNGLUYvRjRGM0Yo JSFH JSFH
<Text-field style="Heading 2" layout="Heading 2">Part b</Text-field> QyQ+SSVkQ2RTRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkiQ0dGJUkiU0dGJSIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqKEkickc2IiIiIiwmRiZGJiokSSJ4R0YlIiIjISIiRiYsJkYmRiZGKUYrRitGJioqLCZJIktHRiVGJioqRiRGJkkiU0dGJSIiJEYpRiZGJ0YmI0YmRjJGJkYxISIkRilGK0YsRishIiM= JSFH
<Text-field style="Heading 2" layout="Heading 2">Part c</Text-field> QyQ+SSdvcHRpbWFHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkPCQvSSVkQ2RTR0YlIiIhL0klZENkeEdGJUYvPCRJIlNHRiVJInhHRiUiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder PCQvSSJTRzYiLCQtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmKiZJI19aR0YpIiIkSSJyR0YlIiIiIiIlSSJLR0YlISIkRjAvSSJ4R0YlI0YyRjA= QyQtSSphbGx2YWx1ZXNHNiI2I0knb3B0aW1hR0YlIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NiU8JC9JIlNHNiIsJCooIiIkIyIiIkYpIiIlIyIiI0YpKiZJIktHRiZGK0kickdGJiEiIkYqI0YpRiwvSSJ4R0YmRio8JC9GJSwkKipGKUYqRixGLUYvRipGMkYtRjNGNDwkL0YlLCQqKkYpRipGLEYtRi9GKkYyRiojISIkRixGNA== Inspect carefully, one solution set with is real, the other terms have complex factors 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 and 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. We can also solve the partial derivative eqations "by hand". Set the derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSVkQ2RTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= equal to 0, then clear denominators. LyoqSSJyRzYiIiIiSSJTR0YlIiIkSSJ4R0YlRiYsJkYmRiYqJEYpIiIjISIiRiYtSSIqRyUqcHJvdGVjdGVkRzYkRiwsJkkiS0dGJUYmRiMjRiZGKA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LyoqSSJyRzYiIiIiSSJTR0YlIiIkSSJ4R0YlRiYsJkYmRiYqJEYpIiIjISIiRiYsJkkiS0dGJUYsRiMjRixGKA== LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEuaXNvbGF0ZX5mb3J+S0YnRi9GMkZD LUkoaXNvbGF0ZUc2IjYkLyoqSSJyR0YkIiIiSSJTR0YkIiIkSSJ4R0YkRiksJkYpRikqJEYsIiIjISIiRiksJkkiS0dGJEYvRicjRi9GK0Yy L0kiS0c2IiwkKipJInJHRiQiIiJJIlNHRiQiIiRJInhHRiRGKCwmRihGKCokRisiIiMhIiJGKCNGKCIiJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH Substitute this into the derivative with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kiS0c2IiwkKipJInJHRikiIiJJIlNHRikiIiRJInhHRilGLSwmRi1GLSokRjAiIiMhIiJGLSNGLSIiJ0klZENkeEdGKUYt LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCgqKiwmKihJInJHNiIiIiJJIlNHRiciIiQsJkYoRigqJEkieEdGJyIiIyEiIkYoI0YoRioqKEYmRihGKUYqRi1GLiMhIiNGKkYoRilGM0YtRi8sJkYoRihGLUYvRi9GKCosRiZGKEYpRihGLUYvRitGKEY0Ri8jRi9GLioqRiZGKEYpRihGK0YoRjRGMyNGKEYu LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw== LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwoKiosJiooSSJyR0YnIiIiSSJTR0YnIiIkLCZGLkYuKiRJInhHRiciIiMhIiJGLiNGLkYwKihGLUYuRi9GMEYzRjQjISIjRjBGLkYvRjlGM0Y1LCZGLkYuRjNGNUY1Ri4qLEYtRi5GL0YuRjNGNUYxRi5GOkY1I0Y1RjQqKkYtRi5GL0YuRjFGLkY6RjkjRi5GNC5JKXN5bWJvbGljR0Yn LCQqLCwmSSJ4RzYiIiIkISIiIiIiRilJInJHRiZGKUkiU0dGJkYpRiVGKCwmRihGKUYlRilGKCNGKCIiJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Now this derivative is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv when 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. Note that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjFGJ0Y5Rjk= are not realistic values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, that is, the values LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjFGJ0Y5Rjk= are not in the domain so the denominator is never LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv in the domain. Likewise, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are not LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv. Then substitute 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 into the expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= LUklc3Vic0clKnByb3RlY3RlZEc2JC9JInhHNiIjIiIiIiIkL0kiS0dGKCwkKipJInJHRihGKkkiU0dGKEYrRidGKiwmRipGKiokRiciIiMhIiJGKiNGKiIiJw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder L0kiS0c2IiwkKiZJInJHRiQiIiJJIlNHRiQiIiQjIiIlIiMiKQ== LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNj9RJyZyYXJyO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR0Y3LyUqdW5kZXJsaW5lR0Y3LyUqc3Vic2NyaXB0R0Y3LyUsc3VwZXJzY3JpcHRHRjcvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGNy8lK2V4ZWN1dGFibGVHRjcvJSlyZWFkb25seUdGNy8lKWNvbXBvc2VkR0Y3LyUqY29udmVydGVkR0Y3LyUraW1zZWxlY3RlZEdGNy8lLHBsYWNlaG9sZGVyR0Y3LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y3LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmdvLUkmbXRleHRHRiQ2NlEuaXNvbGF0ZX5mb3J+U0YnRi9GMkY1RjhGOkY8Rj5GQEZDRkZGSEZKRkxGTkZQRlJGVEZWRllGY28= LUkoaXNvbGF0ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkL0kiS0dGJywkKiZJInJHRiciIiJJIlNHRiciIiQjIiIlIiMiKUYv L0kiU0c2Ii1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCYqJkkjX1pHRiciIiRJInJHRiQiIiIiIiVJIktHRiQhIyIp JSFH QyQtSSphbGx2YWx1ZXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiJUdGKCIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NiUvSSJTRzYiLCQqKCIjIikjIiIiIiIkIiIlIyIiI0YrKiZJIktHRiVGKkkickdGJSEiIkYpI0YqRiwvRiQsJCoqRihGKUYsRi1GL0YpRjJGLUYzL0YkLCQqKkYoRilGLEYtRi9GKUYyRikjRjJGLA== JSFH
JSFH JSFH