Second Derivative Test

Question

Use the Second Derivative Test, if possible, to nd where any local extrema of f(x)

occur. If it is not possible using this test, explain why.

f(x) = x³e^-x/2

Answer 1

f(x) = x³e^-x/2;

f'(x) = 3x²e^-x/2 - x³/2e^-x/2 = x²e^-x/2(3 - x/2) = 0; x = 0 or 3 - x/2 = 0; x = 6

f''(x) = 2xe^-x/2(3 - x/2) - x²/2e^-x/2(3 - x/2) - x²/2e^-x = e^-x/2(6x - x² - 3x²/2 + x³/4 - x²/2) = e^-x/2(6x - 3x² + x³/4)

Now f''(6) = e^-3(36 - 108 + 54) = -18e^-3 < 0

So, by 2nd derivative test we have: at x = 6 we have local maximum f_max = f(6) = 6³e^-3 = 10.754

Now , f''(0) = 0 and we can't use here 2nd derivative test