Question

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

• A. 1
• B. 0
• C. $-\sqrt{3}$
• D. -1

Answer 1

The correct option is D -1

Given , at $x=a,\frac{dy}{dx}=\tan \frac{\pi }{3}=\sqrt{3}$
$\Rightarrow f^{\prime }\left(a\right)=\sqrt{3}$
Also, at $x=b,\frac{dy}{dx}=\tan \frac{\pi }{4}=1$
$\Rightarrow f^{\prime }\left(b\right)=1$
Now, ${\int }_a^b{f}^{\prime }\left(x\right).f^{\prime \prime }\left(x\right)dx$
Put $f^{\prime }\left(x\right)=t\Rightarrow f^{\prime \prime }\left(x\right)dx=dt$
$={\int }_{\sqrt{3}}^{1}tdt$
$=-{\int }_1^{\sqrt{3}}tdt$
$=-[\frac{t^2}{2}{]}_1^{\sqrt{3}}$
$=-1$