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lakkis [162]
3 years ago
8

Mited

Mathematics
1 answer:
Natasha_Volkova [10]3 years ago
7 0

9514 1404 393

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

Step-by-step 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)

__

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

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