# How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for f(x)=x^3+5x^2?

Feb 21, 2018

$f \left(x\right)$ is increasing for $x \in \left(- \infty , - \frac{10}{3}\right) \cup \left(0 , + \infty\right)$
$f \left(x\right)$ is decreasing for $x \in \left(- \frac{10}{3} , 0\right)$
$f \left(x\right)$ has a local maximum at $x = - \frac{10}{3}$ and a local minimum at $x = 0$

#### Explanation:

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

Apply the power rule.

$f ' \left(x\right) = 3 {x}^{2} + 10 x$

$f \left(x\right)$ will be increasing where $f ' \left(x\right) > 0$
and decreasing where $f ' \left(x\right) < 0$

Hence, $f \left(x\right)$ is increasing where:
$3 {x}^{2} + 10 x > 0$

$x \left(3 x + 10\right) > 0 \to$
$x < 0 \mathmr{and} \left(3 x + 10\right) < 0 \mathmr{and} x > 0 \mathmr{and} \left(3 x + 10\right) > 0$
Note: If $x > 0$ then $\left(3 x + 10\right)$ must be $> 0$

$x < 0 \mathmr{and} \left(3 x + 10\right) < 0 \to - \infty < x < - \frac{10}{3}$
and
$x > 0 \to 0 < x < + \infty$

Hence $f \left(x\right)$ is increasing for $x \in \left(- \infty , - \frac{10}{3}\right) \cup \left(0 , + \infty\right)$

$f \left(x\right)$ is decreasing where:
$3 {x}^{2} + 10 x < 0$

$x \left(3 x + 10\right) < 0 \to$
$x < 0 \mathmr{and} \left(3 x + 10\right) > 0 \mathmr{and} x > 0 \mathmr{and} \left(3 x + 10\right) < 0$
Note: If $x > 0$ then $\left(3 x + 10\right)$ cannot be $< 0$

$\therefore x < 0 \mathmr{and} \left(3 x + 10\right) > 0 \to - \frac{10}{3} < x < 0$

Hence, $f \left(x\right)$ is decreasing for $x \in \left(- \frac{10}{3} , 0\right)$

fx) will have turning points where $f ' \left(x\right) = 0$

I.e. where: $x \left(3 x + 10\right) = 0$

$\therefore x = - \frac{10}{3} \mathmr{and} 0$

Since $f \left(x\right)$ is decreasing for $x \in \left(- \frac{10}{3} , 0\right)$ and increasing thereafter, it is clear that:

$f \left(- \frac{10}{3}\right) = {f}_{\max} \mathmr{and} f \left(0\right) = {f}_{\min}$
(This can also be verified by the 2nd derivative test if required)

Hence, $f \left(x\right)$ has a local maximum at $x = - \frac{10}{3}$ and a local minimum at $x = 0$

These results can be observed from the graph of $f \left(x\right)$ below.

graph{x^3+5x^2 [-41.1, 41.1, -20.53, 20.58]}