Let $f, g$  and $h$ be real valued functions defined on the interval $[0,1]$ by  $f\left(x\right)=e^{x^2}+e^{-x^2}$, ,$g\left(x\right)=x\cdot e^{x^2}+e^{-x^2},$ $h\left(x\right)=x^2\cdot e^{x^2}+e^{-x^2}$.  If a, b and c denotes respectively, the absolute maximum of $f$ , $g$ and h on $[0, 1]$  then

$f^1\left(x\right)=2x{e}^{x^2}-2x\cdot e^{-x^2}=2x[e^{x^2}-e^{-x^2}]\ge 0\mathrm{\forall }x\in [0,1]$

$g^1\left(x\right)=e^{x^2}+2x^2\cdot e^{x^2}-2x\cdot e^{-x^2}>0\mathrm{\forall }x\in [0,1]$

$h^1\left(x\right)>0\mathrm{\forall }x\in [0,1]$  
hence  $f, g$  and $h$ are increasing functions in $[0,1]$ , 

Maximum of$f=f\left(1\right)$, and that of  $g\left(x\right)$  and $h\left(x\right)$ and $g\left(1\right)$ and $h\left(1\right),$ 
  $f\left(1\right)=g\left(1\right)=h\left(1\right)=e+\frac{1}{e}\Rightarrow a=b=c$