Find the points of local extrema and local extrema for the function $f\left(x\right)=\cos 4x$ defined on $\left(0,\frac{\pi }{2}\right).$

Given that $f\left(x\right)=\cos 4x;f^{\mathrm{\prime }}\left(x\right)=-4\sin 4x$

$f^{\prime \prime }\left(x\right)=-16\cos 4x$

for Maxima & Minima $f^{\mathrm{\prime }}\left(x\right)=0$

$-4\sin 4x=0$

$4x=0,\pi ,2\pi ,3\pi ,4\pi \dots .$

$x=0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi \dots \dots$

$f^{\prime \prime }\left(\frac{\pi }{4}\right)=-16\cos \left(\frac{4\pi }{4}\right)$

$=-16\cos \pi =-16(-1)=16>0$

The function $f\left(x\right)$ has local minimum at $x=\frac{\pi }{4}$

and its local minimum value is $f\left(\frac{\pi }{4}\right)=-1$.