# How do you find the exact relative maximum and minimum of the polynomial function of g(x) = x^3 - 3x^2 - 9x +1?

Mar 16, 2016

${P}_{\text{max}} \to \left(x , y\right) \to \left(- 1 , 6\right)$
${P}_{\text{min}} \to \left(x , y\right) \to \left(3 , - 26\right)$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {x}^{2} - 6 x - 9 = 0$

Divide throughout by 3 giving

${x}^{2} - 2 x - 3 = 0$

$\left(x + 1\right) \left(x - 3\right) = 0$

$x = - 1 \text{ or } + 3$
'~~~~~~~~~~~~~~~~~~~~~~
$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 6 x - 6$

At x=-1 ; (d^2y)/(dx^2)<0 => "maximum"

At x=+3 ; (d^2y)/(dx^2)>0 => "minimum"

By substitution you find

${P}_{\text{max}} \to \left(x , y\right) \to \left(- 1 , 6\right)$
${P}_{\text{min}} \to \left(x , y\right) \to \left(3 , - 26\right)$