locate the critical points of the following function. then use the second derivative test to determine…

locate the critical points of the following function. then use the second derivative test to determine whether they correspond to local maxima. test is inconclusive. f(x)=1/4x^4 - 8/3x^3 - 32x^2 + 512x identify all the critical points that are local minima by the second derivative test. select the correct choice below and fill in any answer boxes w. a. x= b. there are no critical points that are local minima according to the second derivative test.

locate the critical points of the following function. then use the second derivative test to determine whether they correspond to local maxima. test is inconclusive. f(x)=1/4x^4 - 8/3x^3 - 32x^2 + 512x identify all the critical points that are local minima by the second derivative test. select the correct choice below and fill in any answer boxes w. a. x= b. there are no critical points that are local minima according to the second derivative test.

Answer

Explanation:

Step1: Find the first - derivative

Differentiate $f(x)=\frac{1}{4}x^{4}-\frac{8}{3}x^{3}-32x^{2}+512x$ using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$. $f'(x)=x^{3}-8x^{2}-64x + 512$

Step2: Set the first - derivative equal to zero

$x^{3}-8x^{2}-64x + 512 = 0$ Factor by grouping: $x^{2}(x - 8)-64(x - 8)=0$ $(x - 8)(x^{2}-64)=0$ $(x - 8)(x - 8)(x + 8)=0$ The critical points are $x=-8$ and $x = 8$.

Step3: Find the second - derivative

Differentiate $f'(x)=x^{3}-8x^{2}-64x + 512$ to get $f''(x)=3x^{2}-16x-64$

Step4: Evaluate the second - derivative at the critical points

For $x=-8$: $f''(-8)=3(-8)^{2}-16(-8)-64$ $=3\times64 + 128-64$ $=192+128 - 64$ $=256>0$ For $x = 8$: $f''(8)=3(8)^{2}-16(8)-64$ $=3\times64-128 - 64$ $=192-128 - 64$ $=0$ (test is inconclusive for $x = 8$)

Answer:

A. $x=-8$