# How do you find the relative extrema for f(x) = x^2(6-x)^3?

Jan 14, 2016

Minimum when $x = 0$, maximum when $x = \frac{12}{5}$

#### Explanation:

Find the critical values of the function. These occur when the derivative equals $0$ or is undefined.

To find the derivative of the function, first find the derivative. Although you could distribute the equation, it's probably easier to use the product rule.

$f ' \left(x\right) = {\left(6 - x\right)}^{3} \frac{d}{\mathrm{dx}} \left[{x}^{2}\right] + {x}^{2} \frac{d}{\mathrm{dx}} \left[{\left(6 - x\right)}^{3}\right]$

Find each derivative (the second requires the chain rule):

$\frac{d}{\mathrm{dx}} \left[{x}^{2}\right] = 2 x$

$\frac{d}{\mathrm{dx}} \left[{\left(6 - x\right)}^{3}\right] = 3 {\left(6 - x\right)}^{2} \cdot \frac{d}{\mathrm{dx}} \left[6 - x\right] = - 3 {\left(6 - x\right)}^{2}$

Plug these back in.

f'(x)=2x(6-x)^3+x^2(-3(6-x)^2))

Factor $x {\left(6 - x\right)}^{2}$ from each term.

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

Simplify.

$f ' \left(x\right) = x {\left(6 - x\right)}^{2} \left(12 - 5 x\right)$

This is never undefined. It is equal to $0$ when $x = 0$, $x = 6$, or $x = \frac{12}{5}$.

We can determine what types of extrema these are using the first derivative test (see how the signs change around the points).

Determining $x = 0$:

When $x < 0$, $f ' \left(x\right) < 0$.
When $0 < x < \frac{12}{5} , f ' \left(x\right) > 0$
Since the derivative changes from decreasing to increasing, there is a minimum at $x = 0$.

Determining $x = \frac{12}{5}$:

When $0 < x < \frac{12}{5} , f ' \left(x\right) > 0$
When $\frac{12}{5} < x < 6 , f ' \left(x\right) < 0$
Since the derivative changes from increasing to decreasing, there is a maximum at $x = \frac{12}{5}$.

Determining $x = 6$:

When $\frac{12}{5} < x < 6 , f ' \left(x\right) < 0$
When $x > 6 , f ' \left(x\right) < 0$
Here, the sign of the second derivative doesn't change. This means it is not an extremum, just a point where graph temporarily flattens.

graph{x^2(6-x)^3 [-3, 9, -100, 300]}