For the function $\displaystyle f(x)=5x^{\frac{2}{5}} e^{x} $ , what is the domain of this function?

Mathematica and Matlab says it is only $\displaystyle x\geq 0 $ .

But our math teacher did a solution where she looked for the stationary points of $\displaystyle f(x)$ . She answered that $\displaystyle x=- \frac{2}{5}$ was a stationary point. But this value is negative!

She claimed that the domain of $\displaystyle f(x)$ was $\displaystyle \mathbb{R}$ .