Question

A function $g$ is defined for $\forall x>0$ such that $g\left(1\right)=1;g^{\prime }\left(x^2\right)=x^3\forall x>0$, then the value of $g\left(4\right)$ equals

• A. $\frac{13}{3}$
• B. $3$
• C. $\frac{67}{5}$
• D. None of these

Answer 1

Answer: (C)

$\frac{67}{5}$

Let $x^2=t\Rightarrow g^{\prime }\left(t\right)=t^{3/2}$
or $g\left(t\right)=\frac{t^{5/2}}{5/2}+C=\frac{2}{5}{t}^{5/2}+C$
$g\left(1\right)=1\Rightarrow C=\frac{3}{5}$
$\Rightarrow g\left(t\right)=\frac{2}{5}{t}^{5/2}+\frac{3}{5}\Rightarrow g\left(4\right)=\frac{67}{5}$