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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?

- $-1 \leq x^{*}<-0.5$
- $-0.5 \leq x^{*}<0$
- $x^{*}=0$
- $0<x^* \leq 0.5$
- $0.5<x^* \leq 1$