Let $f:\left[0,1\right]\to R$ be a double differentiable function with $f\left(0\right)=f\left(1\right)=0$ and $f"\left(x\right)+2f\text{'}\left(x\right)+f\left(x\right)\ge 0$ $\forall x\in \left(0,1\right)$ then 

$\text{ Given }{\left(f\left(x\right)e^x\right)}^{\text{'}\text{'}}{\text{=e}}^x\left(f^{"}\left(x\right)+2f\text{'}\left(x\right)+f\left(x\right)\right)\ge 0, \text{ So, }f\left(x\right)e^x\text{ is concave }$ $and f\left(0\right)=0 and f\left(1\right)=0$

$\therefore f\left(x\right)\le 0\forall x\in \left[0,1\right]$

 $option \left(2\right):if f\left(x\right)e^x has minimum at x=\frac{1}{\sqrt{2}} then {\left(f\left(x\right)e^x\right)}^{\text{'}}=e^x\left(f\left(x\right)+f^{\text{'}}\left(x\right)\right)=0 at x=\frac{1}{\sqrt{2}}$

$and f\left(x\right)+f^{\text{'}}\left(x\right) <0 for x\in \left(0,\frac{1}{\sqrt{2}}\right) and f\left(x\right)+f^{\text{'}}\left(x\right) >0 for x\in \left(\frac{1}{\sqrt{2}},1\right)$

$option\left(3\right): \sin ce f\left(0\right)=0 \Rightarrow g\left(0\right)=e^{0}f\left(0\right)=0 and g\left(g\left(0\right)\right)=0 ,g\left(g\left(g\left(0\right)\right)\right)=0$

$similarly f\left(1\right)=0 \Rightarrow g\left(g\left(g\left(1\right)\right)\right)=0$