# What are the critical values, if any, of f(x) = xe^(2x)?

Apr 13, 2017

Critical Value: $x = - \frac{1}{2}$

#### Explanation:

$f \left(x\right) = x {e}^{2 x}$

By Product Rule,

$f ' \left(x\right) = 1 \cdot {e}^{2 x} + x \cdot 2 {e}^{2 x} = \left(1 + 2 x\right) {e}^{2 x}$

By setting $f ' \left(x\right) = 0$,

$\left(1 + 2 x\right) {e}^{2 x} = 0$

By dividing both sides by ${e}^{2 x}$,

$1 + 2 x = 0$

By subtracting 1 from both sides,

$2 x = - 1$

By dividing both sides by $2$,

$x = - \frac{1}{2}$ (Critical Value)

I hope that this was clear.