Question

3. Find the local maxima and local minima, if any, of the following functions.Find also the local maximum and the local minimum values, as the case may be
$\left(i\right)f\left(x\right)=x^2$
$\left(ii\right)g\left(x\right)=x^3-3x$
$\left(iii\right)h\left(x\right)=sinx+cosx,0<x<\frac{\pi }{2}$
$\left(iv\right)f\left(x\right)=\sin x-\cos x,0<x<2\pi$
$\left(v\right)f\left(x\right)=x^3-6x^2+9x+15$
$\left(vi\right)g\left(x\right)=\frac{x}{2}+\frac{2}{x},x>0$
$\left(vii\right)g\left(x\right)=\frac{1}{x^2+2}$
$\left(viii\right)f\left(x\right)=x\sqrt{1-x},x>0$

Answer 1

$f\left(x\right)=x\sqrt{1-x}....(x>0)$
$\begin{array}{c}1-x\ge 0\\ x\le 1\end{array}$
$\therefore Domain:f\left(x\right):x\in (0,1]$
$f^1\left(x\right)=0$
$\Rightarrow \sqrt{1-x}-\frac{1}{2\sqrt{1-x}}(-1)+\sqrt{1-x}.1$
$\Rightarrow \sqrt{1-x}=\frac{2}{\sqrt{1-x}}$
$\Rightarrow 2(1-x)=x$
$\Rightarrow 2-2x=x$
$\Rightarrow 2=\begin{array}{c}f\left(\frac{2}{3}\right)=\frac{2}{3}.\sqrt{1-\frac{2}{3}}\\ =\frac{2}{3}\sqrt{\frac{1}{3}}=\frac{2}{3\sqrt{3}}\to \end{array}$.
local maxima
local minima