Question

Let $\varphi \left(x\right),h\left(x\right)$ and $g\left(x\right)$ be differentiable functions ${\varphi }^{\prime }\left(x\right)=1+\frac{1}{x},\varphi \left(1\right)=1;h\left(x\right)=e^{x^2}$ and $f\left(x\right)=h\left(x\right).\varphi \left(x\right)\forall x\ge 1$. Let $g\left(x\right)$ be inverse of $f\left(x\right)$.

On the basis of above information answer the following questions:$f\left(2\right)$ equal to

• A. $e^4(2+\ln 2)$
• B. $e^4(4+\ln 2)$
• C. $e^4(2+\ln 4)$
• D. $e^4(4+\ln 4)$

Answer 1

The correct option is A $e^4(2+\ln 2)$

${\varphi }^{\prime }\left(x\right)=1+\frac{1}{x}$

Integrating we get, $\varphi \left(x\right)=x+\ln x+c$

Also $\varphi \left(1\right)=1\Rightarrow c=0\Rightarrow \varphi \left(x\right)=x+\ln x$

$\Rightarrow f\left(x\right)=e^{x^2}(x+\ln x)\therefore f\left(2\right)=e^4(2+\ln 2)$