Question

If the tangent to the graph of the function $y=f\left(x\right)$ makes angles of $\pi /4$ and $\pi /3$ with the $x-axis$ at the points $x=2$ and $x=4$ respectively, then the value of ${\int }_2^4{f}^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)dx=$

• A. $0$
• B. $2$
• C. $3$
• D. $1$

Answer 1

The correct option is D $1$

we know tangent to the curve $y=f\left(x\right)$ at $x=2,x=4$ make angle

$\frac{\pi }{4}$ and $\frac{\pi }{3}$ respectively then we must have

$f^{\prime }\left(x\right)=\tan \theta$ put $x=2,\theta =\pi /4$

$f^{\prime }\left(2\right)=1$ put $x=4,\theta =\pi /3$

$f^{\prime }\left(4\right)=\sqrt{3}$

consider

${\int }_2^4{f}^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)dx$ let

${\int }_1^{\sqrt{3}}tdt={\frac{t^2}{2}|}_1^{\sqrt{3}}$

$f^{\prime }\left(x\right)=t$

$f^{\prime \prime }\left(x\right)dx=dt$

$=\frac{3}{2}-\frac{1}{2}=1$