# How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for  3x^5 - 5x^3?

Aug 28, 2017

Take the first derivative...

#### Explanation:

$f ' \left(x\right) = 15 {x}^{4} - 15 {x}^{2}$

Minima and maxima occur at places where the above equation evaluates to zero. Right away, you should be able to see that $x = 0$ is one of these places.

But where else?

$f ' \left(x\right) = 15 {x}^{2} \left({x}^{2} - 1\right)$

$= 15 {x}^{2} \left(x + 1\right) \left(x - 1\right)$

...which gives you the other points: $+ 1 \mathmr{and} - 1$

To determine whether these might be maxima or minima, you must take the second derivative:

$f ' ' \left(x\right) = 60 {x}^{3} - 30 x$

and evaluate at x = -1, 0, and +1.

$f ' ' \left(- 1\right) = - 60 + 30 = - 30$, which is negative, so $x = - 1$ is a relative maxima.

$f ' ' \left(1\right) = 30$, which is positive, so $x = 1$ is a relative minima.

$f ' ' \left(0\right) = 0$, so this point is neither minima nor maxima.

Finding the regions where the original function is increasing or decreasing requires a little more analysis:

Examine the first derivative equation:

( Eq. 1) $f ' \left(x\right) = 15 {x}^{2} \left(x + 1\right) \left(x - 1\right)$

Note that if x < -1, then the terms x+1 and x-1 are both negative, and term $15 {x}^{2}$ is positive, since any number squared is positive.

Therefore, in the region $x < - 1$, the first derivative evaluates to a positive * negative * negative, and is therefore positive. So the original function is increasing where $x < - 1$.

In the region x > 1, the second derivative is obviously positive, so the function is increasing where $x > 1$.

In the region $- 1 < x < 0$, the $x + 1$ term is positive, and the $x - 1$ term is negative. The first derivative is therefore a positive * negative * positive number, which is negative. The original function is therefore DECREASING in the region $- 1 < x < 0$.

(a similar line of reasoning applies to the region $0 < x < 1$).

Always helps to have a graph of the function to serve as a "sanity check".
graph{3x^5 - 5x^3 [-10, 10, -5, 5]}

GOOD LUCK!