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 $x=\pi$