Question

If $f\left(0\right)=f^{\prime }\left(0\right)=0$ and $f^{\prime \prime }\left(X\right)={\tan }^{2}x$ then $f\left(X\right)$ is

• A. $logsecX-\frac{1}{2}{X}^2$
• B. $logcosX+\frac{X^2}{2}$
• C. $logsecX+\frac{1}{2}{X}^2$
• D. $logcosX-\frac{X^2}{2}$

Answer 1

Answer: (A)

$logsecX-\frac{1}{2}{X}^2$

$f^{\prime \prime }\left(x\right)={\tan }^{2}x$

$f^{\prime }\left(x\right)=\int {\tan }^{2}xdx=\int (se{c}^{2}x-1)dx=tanx-x+c$

$f\left(x\right)=\int (tanx-x+c)dx=logsecx-x^2/2+cx+d$

putting $f\left(0\right)=f^{\prime }\left(0\right)=0$

$c=d=0$

$f\left(x\right)=log\sec x-\frac{x^2}{2}$