How to find the coordinates of the stationary points on the curve y = x^3 – 6x^2 – 36x + 16?

Mar 28, 2017

We have a local maximum at $\left(- 2 , 56\right)$ and a local minimum at $\left(6 , - 200\right)$ and an inflexion point at $\left(2 , - 72\right)$

Explanation:

$y = {x}^{3} - 6 {x}^{2} - 36 x + 16$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {x}^{2} - 12 x - 36$

The critical points are when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

That is,

$3 {x}^{2} - 12 x - 36 = 0$

$2 \left({x}^{2} - 4 x - 12\right) = 0$

$2 \left(x + 2\right) \left(x - 6\right) = 0$

Therefore,

$x = - 2$ and $x = 6$

We build a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{a a a a}$$6$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 2$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 6$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$\frac{\mathrm{dy}}{\mathrm{dx}}$$\textcolor{w h i t e}{a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a}$↗$\textcolor{w h i t e}{a a a a}$↘$\textcolor{w h i t e}{a a a a}$↗

Now, we calculate the second derivative

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 6 x - 12$

We have an inflexion point when, $\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 0$

That is, $x = 2$

We make a second chart

$\textcolor{w h i t e}{a a a a}$$I n t e r v a l$$\textcolor{w h i t e}{a a a a}$]-oo,2[$\textcolor{w h i t e}{a a a a}$]2,+oo[

$\textcolor{w h i t e}{a a a a}$$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2$$\textcolor{w h i t e}{a a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\cap$$\textcolor{w h i t e}{a a a a a a a a}$$\cup$

We have a local maximum at $\left(- 2 , 56\right)$ and a local minimum at $\left(6 , - 200\right)$ and an inflexion point at $\left(2 , - 72\right)$