Let $f, g, h$ be real valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^2}+e^{-x^2}, g(x)=x e^{x^2}+e^{-x^2}$ and $h(x)=x^2 e^{x^2}+e^{-x^2}$. If $a, b$ and $c$ denote respectively, the absolute maximum of $f, g$ and $h$ on $[0,1]$, then
Answer & explanation
Correct answer: option 4
For any $x \in[0,1]$, we have
$x^2 \leq x \leq 1$
$\Rightarrow x^2 e^{x^2} \leq x e^{x^2} \leq e^{x^2}$
$\Rightarrow e^{-x^2}+x^2 e^{x^2} \leq e^{-x^2}+x e^{x^2} \leq e^{-x^2}+e^{x^2}$
$\Rightarrow h(x) \leq g(x) \leq f(x)$
Now, $f(x)=e^{x^2}+e^{-x^2}$
$\Rightarrow f^{\prime}(x)=2 x\left(e^{x^2}-e^{-x^2}\right)>0$ for all $x \in(0,1]$
⇒ f(x) is increasing on (0,1]
⇒ f(1) is the maximum value of f(x) on [0, 1]
$\Rightarrow a=e+e^{-1}$
Also, $f(1)=g(1)=h(1)=e+e^{-1}$
∴ $a=b=c=e+e^{-1}$