Consider the function
\[
f(x)=x e^{|x|}+4 x^{2}
\]
for values of $x$ which lie in the interval $[-1,1]$. In this domain, suppose the function attains the minimum value at $x^{*}$. Which of the following is true?
- $-1 \leq x^{*}<-0.5$
- $-0.5 \leq x^{*}<0$
- $x^{*}=0$
- $0<x^* \leq 0.5$
- $0.5<x^* \leq 1$