# What are the critical points of f(x) = e^x(3x^2 - 2x +5)?

Dec 2, 2015

That function has no critical points.

#### Explanation:

The domain of $f$ is $\left(- \infty , \infty\right)$.

$f ' \left(x\right) = {e}^{x} \left(3 {x}^{2} - 2 x + 5\right) + {e}^{x} \left(6 x - 2\right)$

..$= {e}^{x} \left(3 {x}^{2} + 4 x + 3\right)$

$f ' \left(x\right)$ is never undefined and it is never $0$.

To see this, solve ${e}^{x} \left(3 {x}^{2} + 4 x + 3\right) = 0$

${e}^{x} = 0$ has no solutions and $3 {x}^{2} + 4 x + 3 = 0$ has no real solutions.

Therefore, the function has no critical numbers.