Let $f, g$ and $h$ be real-valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^2}+e^{-x^2}, \quad…
Let $f, g$ and $h$ be real-valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^2}+e^{-x^2}, \quad g(x)=x e^{x^2}+e^{-x^2}$ 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
$a=b$ and $c \neq b$
$a=c$ and $a \neq b$
$a \neq b$ and $c \neq b$
$a=b=c$
Solution
Given function,
$
\begin{aligned}
& f(x)=e^{x^2}+e^{-x^2}, \\
& g(x)=x e^{x^2}+e^{-x^2} \text { and } \\
& h(x)=x^2 e^{x^2}+e^{-x^2} \text { are } \quad \text { strictly }
\end{aligned}
$
increasing on $[0,1]$. Hence, at $x=1$, the given function attains absolute maximum all equal to $e+\frac{1}{e}$.
$
\Rightarrow \quad a=b=c
$