Question

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

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

Answer 1

Answer: (B)

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