Find the absolute maximum and minimum values of the function, if they exist, ove

Question

Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval. Also indicate the X-value at which each extremum occurs.
F(X)=4/3x^3-28x; [-3,3]
Note: Answers should be in radical form. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval. Also indicate the X-value at which each extremum occurs.
F(X)=4/3x^3-28x; [-3,3]
Note: Answers should be in radical form.
F(X)=4/3x^3-28x; [-3,3]
Note: Answers should be in radical form.

Explanation / Answer

Solution:

Max and mins can be found at

1) end points of domain
2) critical points (where derivative = 0)
3) points where derivative does not exist

The end points are -3 and 3. So these are some options.

Critical points:

y = (4/3)x^3 - 28x
dy/dx = 4x^2 - 28
0 = 4x^2 - 28
28 = 4x^2 => x^2 = 7
x = +/- sqrt(7)

There are no points where dy/dx does not exist.

If x = -3, then y = -36 + 84 = 48 < 49
If x = -sqrt(7), then y = (4/3){-sqrt(7)}^3 - 28{-sqrt(7)} = (56/3)sqrt(7) > 49
If x = sqrt(7), then y = (4/3){sqrt(7)}^3 - 28{sqrt(7)} = -(56/3)sqrt(7) < -49
If x = 3, then y = 36 - 84 = -48 > -49

Maximum is (56/3)sqrt(7) at x = -sqrt(7).
Minimum is -(56/3)sqrt(7) at x = sqrt(7)

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