Question

Let $f\left(x\right)=e^x-x$ and $g\left(x\right)x^2-x,\forall x\in R$. Then the set of all $x\in R$, where the function $h\left(x\right)=\left(fog\right)\left(x\right)$ is increasing is:

• A. $[-1,\frac{-1}{2}]\cup [\frac{1}{2},\infty )$
• B. $[0,\frac{1}{2}]\cup [1,\infty )$
• C. $[\frac{-1}{2},0]\cup [1,\infty )$
• D. $[0,\infty )$

Answer 1

Answer: (B)

$[0,\frac{1}{2}]\cup [1,\infty )$

$h\left(x\right)=f\left(g\right(x\left)\right)$
$\Rightarrow h^{\prime }\left(x\right)=f^{\prime }\left(g\right(x\left)\right).g^{\prime }\left(x\right)$ and $f^{\prime }\left(x\right)=e^x-1$
$\Rightarrow h^{\prime }\left(x\right)=(e^{g\left(x\right)}-1)g^{\prime }\left(x\right)$
$\Rightarrow h^{\prime }\left(x\right)=(e^{x^2}-1)(2x-1)\ge 0$
Case -I $e^{x^2-x}\ge 1$ and $2x-1\ge 0$
$\Rightarrow x\in [1,\infty )$ ...(1)
Case - II $e^{x^2-x}\le 1$ and $2x-1\le 0$
$\Rightarrow x\in [0,\frac{1}{2}]$ ...(2)
Hence, $x\in [0,\frac{1}{2}]\cup [1,\infty )$