Find the points of local maxima or minima and corresponding local maximum and minimum values of `f(x)=xsqrt(32-x^2),\ -5lt=xlt=5` . Also, find the points of inflection, if any:

Explanation

Given function: `f(x)=x sqrt{32-x^{2}},-5 leq x leq 5`

Differentiating with respect to `x`

`f^{prime}(x)=sqrt{32-x^{2}}+frac{x}{2 sqrt{32-x^{2}}} times(-2 x)`

`=frac{2(32-x^{2})-2 x^{2}}{2 sqrt{32-x^{2}}}`

`=frac{64-4 x^{2}}{2 sqrt{32-x^{2}}}`

Again differentiating with respect to `x`

`f^{prime prime}(x)=frac{2 sqrt{32-x^{2}}(-8 x) frac{-2(64-4 x^{2})}{2 sqrt{32-x^{2}}}(-2 x)}{2 sqrt{32-x^{2}}}`

`=frac{-4(32-x^{2}) 8 x+4 x(64-x^{2})}{8(32-x^{2})^{frac{3}{2}}}`

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