Question

Let $f(x)=\int\limits_0^x\left(5 \ln \left(1+y^2\right)-10 y \tan ^{-1} y+16 \sin y\right) d y$, then which of the following hold(s) good?

• A. $f^{\prime}(x)$ is increasing for all $x \in(0,1)$.
• B. $f(x)$ is positive for all $x \in(0,1)$.
• C. $f ( x )$ is negative for all $x \in(0,1)$.
• D. $\int\limits_0^x f(y) d y$ is increasing for all $x \in(0,1)$.

Answer 1

Answer: (A), (C), (D)

$ f ^{\prime}( x )=5 \ln \left(1+ x ^2\right)-10 x \tan ^{-1} x +16 \sin x $
$f ^{\prime \prime}( x )=2\left(8 \cos x -5 \tan ^{-1} x \right) $
$f ^{\prime \prime \prime}( x )=-2\left(8 \sin x +\frac{5}{1+ x ^2}\right)<0 \forall x \in(0,1)$
So, $f ^{\prime \prime}( x )$ is decreasing function $\forall x \in(0,1)$
$\therefore f ^{\prime \prime}( x )> f ^{\prime \prime}(1) \Rightarrow f ^{\prime \prime}( x )>0$
$\Rightarrow f^{\prime}(x)$ is increasing function for $x>0$
$f ^{\prime}( x )> f ^{\prime}(0) $
$f ^{\prime}( x )>0 .$
$\Rightarrow f ( x )$ is increasing function for $x >0$
Hence, $f(x)>f(0)$
$\Rightarrow f ( x )>0$
$\therefore \int\limits_0^x f(y) d y \text { is increasing for all } x \in(0,1)$