Question

Find the critical numbers and the intervals on which the function f(x)=9x5−3x3+6f(x)=9x5−3x3+6 is increasing or decreasing. Use the First Derivative Test to determine whether the critical number is a local minimum or maximum (or neither). (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. Enter NONE if there are no critical points.) The critical numbers with local minimum : help (fractions) The critical numbers with local maximum : (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (*,*). Use inf for infinity, U for combining intervals, and appropriate type of parenthesis "(", ")", "[" or "]" depending on whether the interval is open or closed.) The function is increasing on help (intervals) The function is decreasing on

Answer 1

x = (0, -√(3/5), √(3/5)) are the critical points.

Local maximum at √(3/5), and local minimum is at -√(3/5).

The function is increasing in the interval

(−∞, -√(3/5)) U (√(3/5), ∞)

And decreasing in the interval

(-√(3/5), 0) U (0, √(3/5))

Explanation:

Given the function

f(x) = 9x^5 - 3x³ + 6

First of all, take the first derivative of this, to have

f'(x) = 45x^4 - 27x²

The critical point are the points where the first derivative vanishes, that is

f'(x) = 0

Now, solve the equation

45x^4 - 27x² = 0

9x²(5x² - 3) = 0

x = 0 twice

Or

5x² - 3 = 0

5x² = 3

x² = 3/5

x = ±√(3/5)

So, x = (0, -√(3/5), √(3/5)) are the critical points.

Local maximum is when f'(x) > 0, this is √(3/5) in this case,

and local minimum is when f'(x) < 0, this is -√(3/5) in this case.

Now, we need to test for the various intervals to determine where the function increases and decreases.

(−∞, -√(3/5)):

f'(-√(4/5)) = 45(-√(4/5))^4 - 27(-√(4/5))²

= 36/5 > 0. Increasing

(-√(3/5), 0):

f'(-√(2/5)) = 45(-√(2/5))^4 - 9(-√(2/5))²

= -18/5 < 0. Decreasing

(0, √(3/5)):

f'(√(1/5)) = 45(√(1/5))^4 - 9(√(1/5))² = -18/5 < 0. Decreasing

(√(3/5), ∞): f'(1) = 45(1)^4 - 9(1)² =

36 > 0. Increasing.

Therefore, the function is increasing in the interval

(−∞, -√(3/5)) U (√(3/5), ∞)

And decreasing in the interval

(-√(3/5), 0) U (0, √(3/5))