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

1 Answer
Aug 28, 2017

Take the first derivative...

Explanation:

f'(x) = 15x^4 - 15x^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'(x) = 15x^2(x^2 - 1)

=15x^2(x+ 1)(x - 1)

...which gives you the other points: +1 and -1

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

f''(x) = 60x^3 - 30x

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

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

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

f''(0) = 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'(x) = 15x^2(x+ 1)(x - 1)

Note that if x < -1, then the terms x+1 and x-1 are both negative, and term 15x^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!