If the tangent to the graph function $y = f(x)$ makes angles $\frac{\pi}{4}$ and $\frac{\pi}{3}$ with the $x$-axis is at the point $x = 2$ and $x = 4$ respectively, the value of $\int_{2}^{4}{f^{'}(x)f^{''}(x)}\ dx$

Question

Dr. Abubacker · Accepted Answer

(d)
Given, $f^{'}(2) = 	an{\frac{\pi}{4} = 1,\ f^{'}(4) = 	an{\frac{\pi}{3} = \sqrt{3}}}$
$	herefore\int_{2}^{4}{f^{'}(x)f^{''}(x)dx = \left\lbrack \frac{1}{2}\lbrack f^{'}(x)\left. \ \  ightbrack^{2} ightbrack_{2}^{4}}$
$= \frac{1}{2}\lbrack f^{'}(4)\left. \ \  ightbrack^{2} - \frac{1}{2}\left\lbrack f^{'}(2) ightbrack^{2}$
$= \frac{3}{2} - \frac{1}{2} = 1$

Q. If the tangent to the graph function $y = f(x)$ makes angles $\frac{\pi}{4}$ and $\frac{\pi}{3}$ with the $x$-axis is at the point $x = 2$ and $x = 4$ respectively, the value of $\int_{2}^{4}{f^{'}(x)f^{''}(x)}\ dx$

Explanation

Reviewed by Dr. Abubacker

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