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

Explanation

Given function: `f(x)=_{x} sqrt{2-x^{2}}`

Differentiating with respect to `x`

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

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

`=frac{2-2 x^{2}}{sqrt{2-x^{2}}}`

Again differentiating with respect to `x`

`f^{prime prime}(x)=frac{sqrt{2 x^{2}}(4 x)+frac{(2-2 x^{2}) 2 x}{sqrt{2-x^{2}}}}{(sqrt{.2-x^{2})^{2}}}`

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

...