Question

For the function $f\left(x\right)={\int }_0^x\frac{sint}{t}dt$ where x > 0

• A. Maximum occurs at $x=n\pi$, n is even
• B. Maximum occurs at $x=n\pi$, n is odd
• C. Minimum occurs at $x=n\pi$ , n is prime number
• D. Minimum occurs at $x=n\pi$, n is odd

Answer 1

The correct option is B Maximum occurs at $x=n\pi$, n is odd

$f^{{}^{\prime }}\left(x\right)=\frac{sinx}{x}and{f}^{\prime \prime }\left(x\right)=\frac{xcosx-sinx}{x^2}$
$f^{{}^{\prime }}\left(x\right)=0\Rightarrow sinx=0andx\ne 0$
$\therefore x=n\pi ,f^{\prime \prime }\left(x\right)=\frac{(-1{)}^n}{n\pi }$
$\therefore$ Max occurs at odd $\pi$ and minimum at even $\pi$