Locate the critical points of the following function. Then use the Second Deriva

Question

Locate the critical points of the following function. Then use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither. f(x) = 4x^2 e^-x - 1 What is(are) the critical point(s) off? Select the correct choice below and. if necessary, fill in the answer box to complete your choice. The critical point(s) is(are) x =. (Use a comma to separate answers as needed.) There are no critical points for f. What is/are the local minimum/minima off? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The local minimum/minima of f is/are at x = (Use a comma to separate answers as needed.) There is no local minimum of f. What is/are the local maximum/maxima off? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The local maximum/maxima of f is/are at x = (Use a comma to separate answers as needed.) There is no local maximum of f.

Explanation / Answer

given f(x)=4x2e-x -1

differentiate with respect to x

f '(x)=8xe-x+4x2(-e-x) -0

f '(x)=(8x-4x2)e-x

f '(x)=4x(2-x)e-x

for critical points f '(x)=0

4x(2-x)e-x=0

4x(2-x)=0

x=0,x=2

x=0,2 are critical points

for x<0 , f '(x)<0, for 0<x<2 ,f '(x)>0

so by first derivative test f(x) has local minimum at x =0

for x<2 , f '(x)>0, for x>2 ,f '(x)<0

so by first derivative test f(x) has local maximum at x =2

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