# The function f(x) = x(64-x^2)^1/2 satisfies the hypotheses of Rolle's Theorem on the interval [-8,8]. Find all values of that satisfy the conclusion o

The function $f\left(x\right)=x\frac{{\left(64-{x}^{2}\right)}^{1}}{2}$ satisfies the hypotheses of Rolles
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2abehn
Your derivative should have been $\sqrt{64-{x}^{2}}-\frac{{x}^{2}}{\sqrt{64-{x}^{2}}}$
The minus sign coming from the derivative of $\left(-{x}^{2}\right)$
Setting the derivative to zero,
$\sqrt{64-{x}^{2}}-\frac{{x}^{2}}{\sqrt{64-{x}^{2}}}=0$ multiply both terms by $\sqrt{64-{x}^{2}}$
$\left(64-{x}^{2}\right)-{x}^{2}=0$
$64=2{x}^{2}$
$32={x}^{2}$
$x=\frac{+}{-}4.\sqrt{2}$