# How do you find the local maximum and minimum values of g(x)=x^3+5x^2-17x-21?

Local Maximum : $\left(- 4.57259929569 , 65.6705658828\right)$
Local Minimum : $\left(1.2392659623 , - 32.48538069\right)$

#### Explanation:

From the given equation
$y = {x}^{3} + 5 {x}^{2} - 17 x - 21$

take the first derivative

$y ' = \frac{d}{\mathrm{dx}} \left({x}^{3}\right) + \frac{d}{\mathrm{dx}} \left(5 {x}^{2}\right) + \frac{d}{\mathrm{dx}} \left(- 17 x\right) + \frac{d}{\mathrm{dx}} \left(- 21\right)$

$y ' = 3 {x}^{2} + 10 x - 17$

Set $y ' = 0$ then solve for x

$3 {x}^{2} + 10 x - 17 = 0$

$x = \frac{- 10 \pm \sqrt{{\left(10\right)}^{2} - 4 \left(6\right) \left(- 17\right)}}{6}$

$x = \frac{- 10 \pm \sqrt{304}}{6}$

to values for x:

${x}_{1} = - 4.57259929569$
and

${x}_{2} = 1.2392659623$

Solve for corresponding values of y using $y = {x}^{3} + 5 {x}^{2} - 17 x - 21$ and the points are

Local Maximum: $\left(- 4.57259929569 , 65.6705658828\right)$
Local Minimum : $\left(1.2392659623 , - 32.48538069\right)$

Kindly see the graph below for better view

God bless....I hope the explanation is useful.