# How do you find the local maximum and minimum values of f(x) = 5 + 9x^2 − 6x^3 using both the First and Second Derivative Tests?

Aug 4, 2017

$\left(0 , 5\right) \setminus \setminus \setminus \setminus \setminus \setminus \setminus =$ minimum
$\left(1 , 8\right) \setminus \setminus \setminus \setminus \setminus \setminus \setminus =$ maximum
$\left(\frac{1}{2} , \frac{13}{2}\right) =$ non-stationary inflection point

#### Explanation:

We have:

$f \left(x\right) = 5 + 9 {x}^{2} - 6 {x}^{3}$

We can see the critical point via a graph:

graph{5 + 9x^2-6x^3 [-6, 6, -2, 14]}

We can examine the critical points using calculus:

Differentiating wrt $x$ we get:

$f ' \left(x\right) = 18 x - 18 {x}^{2}$

At a critical point we have $f ' \left(x\right) = 0$

$f ' \left(x\right) = 0 \implies 18 x - 18 {x}^{2} = 0$

$\therefore 18 x \left(1 - x\right) = 0 \implies x = 0 , 1$

And, now we have the $x$-coordinates, we can determine the nature of the turning points (or critical points) by using the second derivative test. Differentiating a second time, we get:

$f ' ' \left(x\right) = 18 - 36 x$

When:

$x = 0 \implies f ' ' \left(0\right) = 18 - 0 \setminus \setminus > 0 \implies$minimum
$x = 1 \implies f ' ' \left(1\right) = 18 - 36 < 0 \implies$maximum

Also note we have an inflection point if $f ' ' \left(x\right) = 0$

$f ' ' \left(x\right) = 0 \implies 18 - 36 x = 0 \implies x = \frac{1}{2}$

Now we have the $x$-coordinate of the critical points let us find the associated $y$-coordinate:

$x = 0 \setminus \implies f \left(0\right) \setminus \setminus \setminus \setminus \setminus = 5 + 0 - 0 = 5$
$x = 1 \setminus \implies f \left(1\right) \setminus \setminus \setminus \setminus \setminus = 5 + 9 - 6 = 8$
$x = \frac{1}{2} \implies f \left(\frac{1}{2}\right) = 5 + 9 \left(\frac{1}{4}\right) - 6 \left(\frac{1}{8}\right) = \frac{13}{2}$

Hence, in summary

$\left(0 , 5\right) \setminus \setminus \setminus \setminus \setminus \setminus \setminus =$ minimum
$\left(1 , 8\right) \setminus \setminus \setminus \setminus \setminus \setminus \setminus =$ maximum
$\left(\frac{1}{2} , \frac{13}{2}\right) =$ non-stationary inflection point

Which is consistent with what we see graphically

Aug 4, 2017

The local maximum is $= \left(1 , 8\right)$ and the local minimum is $= \left(0 , 5\right)$
The point of inflection is $= \left(\frac{1}{2} , \frac{13}{2}\right)$

#### Explanation:

Our function is

$f \left(x\right) = 5 + 9 {x}^{2} - 6 {x}^{3}$

The first derivative is

$f ' \left(x\right) = 18 x - 18 {x}^{2}$

The critical points are when

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

$18 x - 18 {x}^{2} = 18 x \left(1 - x\right)$

$18 x \left(1 - x\right) = 0$

Therefore, $x = 0$ and $x = 1$

We can build a variation 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}$$0$$\textcolor{w h i t e}{a a a a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\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}$$-$$\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}$$1 - x$$\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}$$f ' \left(x\right)$$\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}$$f \left(x\right)$$\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}$↘

Now, we calculate the second derivative

$f ' ' \left(x\right) = 18 - 36 x$

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

That is,

$18 - 36 x = 0$, $\implies$, $x = \frac{18}{36} = \frac{1}{2}$

We build a variation chart with the second derivative

$\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}$$\left(- \infty , \frac{1}{2}\right)$$\textcolor{w h i t e}{a a a a}$$\left(\frac{1}{2} , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$s i g n f ' ' \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a 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}$$\cup$$\textcolor{w h i t e}{a a a a a a a a a a a a}$$\cap$

graph{5+9x^2-6x^3 [-15.35, 16.69, -3.14, 12.88]}