All of this is wrong. Here is the actual question: "Find the local maximum and minimum values offusing both the First and Second Derivative Tests." Please look thru this to see what is mistaken.

$\displaystyle f(x) = -8x^{3} + 12x^{2} + 8$

$\displaystyle f'(x) = -24x^{2} + 24x$

$\displaystyle f''(x) = -48x + 24$

Find Critical Numbers:

$\displaystyle f'(x) = -24x^{2} + 24x$

$\displaystyle -24x^{2} + 24x = 0$

$\displaystyle -24x^{2} = - 24$

$\displaystyle x^{2} = 1$

$\displaystyle \sqrt{x^{2}} = \sqrt{1}$

$\displaystyle x = \pm \sqrt{1}$

$\displaystyle x = -1$

$\displaystyle x = 1$

Find Min Max

$\displaystyle f(x) = -8x^{3} + 12x^{2} + 8$

$\displaystyle f(-1) = -8(-1)^{3} + 12(-1)^{2} + 8 = 28$

$\displaystyle f(1) = -8(1)^{3} + 12(1)^{2} + 8 = 12$