# What are the values and types of the critical points, if any, of f(x)=x^3 + 3x^2 + 1?

Apr 16, 2017

We have a local maximum at $\left(- 2 , 5\right)$
We have a local minimum at $\left(0 , 1\right)$
The inflexion point is $\left(- 1 , 3\right)$

#### Explanation:

We calculate the first derivative

$f \left(x\right) = {x}^{3} + 3 {x}^{2} + 1$

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

The critical points are when, $f ' \left(x\right) = 0$

$3 {x}^{2} + 6 x = 0$

Factorising yields

$3 x \left(x + 2\right) = 0$

Therefore,

$x = 0 \mathmr{and} x = - 2$

We build a 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}$$0$$\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}$$x$$\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}$$+$

$\textcolor{w h i t e}{a a a a}$$f ' \left(x\right)$$\textcolor{w h i t e}{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}$$f \left(x\right)$$\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}$↗

Now, we calculate the second derivative

$f ' ' \left(x\right) = 6 x + 6$

The inflexion point is when $f ' ' \left(x\right) = 0$

$6 x + 6 = 0$

$x = - 1$

The inflexion point is $\left(- 1 , 3\right)$

We calculate

$f ' ' \left(- 2\right) = 6 \cdot - 2 + 6 = - 6$

As $f ' ' \left(- 2\right) < 0$, we have a local maximum at $\left(- 2 , 5\right)$

$f ' ' \left(0\right) = 6$

As $f ' ' \left(0\right) > 0$, we have a local minimum at $\left(0 , 1\right)$

We build another chart to determine the convexity and concavity

$\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}$$\left(- \infty , - 1\right)$$\textcolor{w h i t e}{a a a a}$$\left(- 1 , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$f ' ' \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\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 a a a}$$\cup$

graph{x^3+3x^2+1 [-14.66, 13.82, -5.92, 8.32]}