Question

Mited

Find any relative extrema of the function. List each extremum along with the x-value at which it occurs. Identify intervals over which the function is
increasing and over which it is decreasing. Then sketch a graph of the function.
f(x) = -x^3+ 9x?

Answer 1

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Answer:

• relative minimum -6√3 at x = -√3
• relative maximum 6√3 at x = √3
• decreasing on x < -√3 and x > √3
• increasing on -√3 < x < √3
• see below for a graph

Explanation:

I find it convenient to draw the graph first when looking for relative extrema.

The function can be differentiated to get ...

f'(x) = -3x^2 +9

This is zero when ...

-3x^2 +9 = 0

x^2 = 3

x = ±√3 . . . . . x-values of relative extrema

Then the extreme values are ...

f(±√3) = x(9 -x^2) = (±√3)(9 -3) = ±6√3

The lower extreme (minimum) corresponds to the lower value of x (-√3), so the extrema are ...

(x, y) = (-√3, -6√3) and (√3, 6√3)

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Since the leading coefficient is negative and the degree is odd, the function is decreasing for values of x below the minimum and above the maximum. It is increasing for values of x between the minimum and the maximum.

decreasing: x < -√3, and √3 < x

increasing: -√3 < x < √3