Find the critical numbers and the intervals on which the function f(x)=8x^55x^3+

Question

Find the critical numbers and the intervals on which the function f(x)=8x^55x^3+9 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 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?

The function is decreasing on?

Explanation / Answer

(a) f(x) = 8x^5 5x^3 + 9

f'(x) = 40x^4 - 15x^2 = 5x^2 (8x^2 - 3)

f'(x) = 0 gives 5x^2 (8x^2 - 3) = 0

On solving, we get x = 0, -3 /(2 2), 3 /(2 2), that is x = 0, -0.6124, 0.6124 are the critical numbers

f'(x < 0) and f'(x > 0) are both positive. So x = 0 is neither a point of local maximum nor a point of local minimum

f'(x < -0.6124) is positive and f(x > -0.6124) is negative. This means x = -0.6124 is a point of local maximum.

f'(x < 0.6124) is negative and f(x > 0.6124) is positive. This means x = 0.6124 is a point of local minimum..

(b) f'(x) > 0 implies 8x^2 - 3 > 0. On solving this inequality we get x < -0.6124 or x > 0.6124

So the function is increasing in (-, -0.6124) U (0.6124, )

f'(x) < 0 implies 8x^2 - 3 < 0. On solving this inequality we get -0.6124 < x < 0.6124

So the function is decreasing in (-0.6124, 0.6124)

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