Question

Prove that the function given by $f\left(x\right)=\cos x$ is neither increasing nor decreasing in $(0,2\pi )$.

Answer 1

$f\left(x\right)=\cos x\Rightarrow f^{{}^{\prime }}\left(x\right)=-\sin x$

$f^{{}^{\prime }}\left(x\right)$ is $+ve$ for $xϵ(\pi ,2\pi )$ & $f^{{}^{\prime }}\left(x\right)$ is $-ve$ for $xϵ(0,\pi )$.

Hence over entire interval $xϵ(0,2\pi ),f\left(x\right)$ is neither increasing, nor decreasing.