Question

Consider the function
$f\left(x\right)=x\cos x-\sin x$ 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

The correct option is C f has a maxima at $x=\pi$

$f\left(x\right)=x\cos x-\sin x$
$f^{\prime }\left(x\right)=-x\sin x$
$f^{\prime }\left(x\right)=0\Rightarrow x=0,\pi$
$f^{\prime \prime }\left(x\right)=-x\cos x-\sin x$
$f^{\prime \prime }\left(\pi \right)=-\pi <0$
Hence, $f$ has a maxima at $x=\pi$