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for what values of c does the curve y = c(x^3) + e^x have inflection points? Sol

ID: 2972680 • Letter: F

Question

for what values of c does the curve y = c(x^3) + e^x have inflection points?

Explanation / Answer

In this example dy/dx = 3cx^2 + e^x ----> d2y/dx2 = 6cx + e^x We are therefore looking for real solutions to 6cx + e^x = 0 ----> e^x = -6cx Consider the curve y = e^x and the line y = -6cx. Can they intersect? They can if c is any positive value. A sketch makes this obvious. They can also intersect if c is negative and large enough. The line through the origin which is a tangent to y = e^x is y = e intersecting at (1, e) Therefore you need -6c >= e ----> c =< -e/6 Full solution (-inf

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