Question

Consider a function $f\left(x\right)=\frac{sinx}{2}$. Let $g\left(x\right)=\int f\left(x\right)dx$, where constant of integration is zero.
On the basis of above information, answer the following questions The number of local minima of $g\left(x\right)$ in (2$\pi$,12$\pi$) are

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

The correct option is B $5$

Given,

$f\left(x\right)=\frac{\sin x}{2}$

$g\left(x\right)=\int f\left(x\right)dx$

$=\int \frac{\sin x}{2}dx$

$=-\frac{1}{2}\cos x+c$

$c=0$ (given in question

Now for critical points,

$g^{\prime }\left(x\right)=0$

$-\frac{1}{2}\times (-\sin x)=0$

$\sin x=0$

$x=n\pi (n=0,1,\mathrm{2...})$

For maxima/minima

$g^{\prime \prime }\left(x\right)=\cos x$ [ positive for $(0,\frac{\pi }{2}),(x,\frac{3\pi }{2})...$]

We have to consider positive values for minima

Between $(2\pi ,12\pi )$ there will be total $10$ critical points out of which $5$ points will give minima.

$\left(5\right)$ is the correct answer.