Question

A line tangent to the graph of the function $y=f\left(x\right)$ at the point $x=a$ forms an angle $\pi /3$ with $y$-axis and at $x=b$ and angle $\pi /4$ with $x$-axis then ${\int }_a^{b}f{}^{\prime \prime }\left(x\right)dx$ is

• A. $1/\sqrt{3}-1$
• B. $-\pi /12$
• C. $\pi /12$
• D. $\sqrt{3}-1$

Answer 1

Answer: (A)

$1/\sqrt{3}-1$

$f^{\prime }\left(a\right)=\tan (\frac{\pi }{2}-\frac{\pi }{3})=\cot \frac{\pi }{3}=\frac{1}{\sqrt{3}}$
and $f^{\prime }\left(b\right)=\tan \frac{\pi }{4}=1$
So,
${\int }_a^b{f}^{\prime \prime }\left(x\right)dx=f^{\prime }\left(b\right)-f^{\prime }\left(a\right)=\frac{1}{\sqrt{3}}-1$