Let $f$, $g$ and $h$ 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

$f(x) = e^{x^2} + e^{-x^2}$

$g(x) = x\,e^{x^2} + e^{-x^2}$

$h(x) = x^2 e^{x^2} + e^{-x^2}$

$a = \max_{[0,1]} f, \quad b = \max_{[0,1]} g, \quad c = \max_{[0,1]} h$

$\text{Find the relation among } a, b, c$

$1.\ \text{plot } f, g, h \text{ on } [0,1]$

$2.\ \text{sign of } f', g', h' \Rightarrow \text{increasing}$

$3.\ \text{max at } x = 1 \text{ (closed interval)}$

$f'(x) = 2x\,e^{x^2} - 2x\,e^{-x^2} = 2x\left(e^{x^2} - e^{-x^2}\right)$

$x \in [0,1]: \ f'(x) \ge 0 \Rightarrow f \text{ increasing, so } a = f(1)$

$g'(x) = e^{x^2}(1 + 2x^2) - 2x\,e^{-x^2}$

$e^{x^2} g'(x) = e^{2x^2}(1+2x^2) - 2x \ge (1+2x^2) - 2x$

$= 2\left(x - \tfrac12\right)^2 + \tfrac12 > 0 \Rightarrow b = g(1)$

$h'(x) = 2x\,e^{x^2}(1+x^2) - 2x\,e^{-x^2}$

$= 2x\left[e^{x^2}(1+x^2) - e^{-x^2}\right] \ge 0 \Rightarrow c = h(1)$

$f(1) = e + \dfrac1e, \quad g(1) = e + \dfrac1e, \quad h(1) = e + \dfrac1e$

$\Rightarrow a = b = c = e + \dfrac1e$

$a = b = c \Rightarrow \text{option (4)}$