Question

Find the point of local maxima & local minima of the function
$f\left(x\right)={\sin }^{4}x+{\cos }^{4}xin\left[0,\pi \right]$.

Answer 1

$f\left(x\right)={\sin }^{4}x+{\sin }^{4}x$

$={\left({\sin }^{2}x+{\cos }^{2}x\right)}^2-2{\sin }^{2}x{\cos }^{2}x$

$=1-\frac{{\sin }^{2}2x}{2}$

f(x) will be minimum when

${\sin }^{2}2x=1$

$\sin 2x=\pm 1$

$2x=\frac{\pi }{2}$

$\Rightarrow x=\frac{\pi }{4}$

f(x) will be maximum when

${\sin }^{2}2x=0$

$2x=n\pi$

$x=\frac{\pi }{2}$

and , $x=\frac{\pi }{4}$

Answer: $x=0,\frac{\pi }{4},\frac{\pi }{2}$