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!