# How do you find the critical points for f(x)=8x^3+2x^2-5x+3?

Feb 23, 2015

Hello,

Calculate the derivative

$f ' \left(x\right) = 24 {x}^{2} + 4 x - 5$.

Solve $f ' \left(x\right) = 0$. To do that, Calculate the discriminant $\Delta = {4}^{2} - 4 \setminus \times 24 \times \left(- 5\right) = 496$.

The critical points of $f$ are the zeros of $f '$. So they are
$\frac{- 4 - \sqrt{496}}{48}$ and $\frac{- 4 + \sqrt{496}}{48}$.

You can simplify, because $496 = 16 \setminus \times 31$ :
$\frac{- 1 - \sqrt{31}}{12}$ and $\frac{- 1 + \sqrt{31}}{12}$.