Question

$f\left(x\right)=\{\begin{array}{cc}|x^3+x^2+3x+\sin x|\cdot (3+\sin \frac{1}{x}),& x\ne 0\\ 0,& x=0\end{array}$. The number of points, where $f\left(x\right)$ attains its minimum value, is

• A. $1$
• B. $2$
• C. $3$
• D. Infinitely many

Answer 1

Answer: (A)

$1$

$f\left(x\right)=\{\begin{array}{cc}|x^3+x^2+3x+\sin x|\cdot (3+\sin \frac{1}{x}),& x\ne 0\\ 0,& x=0\end{array}$
Let $g\left(x\right)=x^3+x^2+3x+\sin x$
$g^{\prime }\left(x\right)=3x^2+2x+3+\cos x$
$=3(x^2+\frac{2x}{3}+1)+\cos x$
$\Rightarrow g^{\prime }\left(x\right)=3\{{(x+\frac{1}{3})}^2+\frac{8}{9}\}+\cos x>0$
and $2<3+\sin \left(\frac{1}{x}\right)<4$
Hence, minimum value of $f\left(x\right)$ is $0$ at $x=0$
Hence, the number of points $=1$.