Question

Consider the function $f(x)=x \cos x-\sin x$, then identify the statement which is correct .

• A. $f$ is neither odd nor even
• B. f is monotonic decreasing at $x=0$
• C. f has a maxima at $x=\pi$
• D. $f$ has a minima at $x=-\pi$

Answer 1

Answer: (B)

$f^{\prime}(x)=-x \sin x=0$ when $x=0$ or $\pi,-\pi$

This also implies that $f$ is decreasing at $x =0 \Rightarrow( B )$ is correct $f^{\prime \prime}(x)=-(x \cos x+\sin x)$
$f ^{\prime \prime}(\pi)=-(-\pi)>0 $ minima at $x=\pi$
$f ^{\prime \prime}(-\pi)=-(\pi)<0 $ maxima at $x=\pi $