# How do you find the absolute maximum and absolute minimum values of f on the given interval #f(x) = x^5 - x^3 - 1# and [-1, 1]?

Absolute minimum

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To find the absolute maximum and minimum values of ( f(x) = x^5 - x^3 - 1 ) on the interval ([-1, 1]), we follow these steps:

- Find the critical points of ( f(x) ) in the interval by setting its derivative equal to zero and solving for ( x ).
- Evaluate ( f(x) ) at these critical points and at the endpoints of the interval.
- The largest value obtained from these evaluations is the absolute maximum, and the smallest value is the absolute minimum.

First, we find the derivative of ( f(x) ): [ f'(x) = 5x^4 - 3x^2 ]

Next, we find the critical points by setting ( f'(x) = 0 ): [ 5x^4 - 3x^2 = 0 ] [ x^2(5x^2 - 3) = 0 ] [ x = 0 \text{ or } x = \pm \sqrt{\frac{3}{5}} ]

Now, we evaluate ( f(x) ) at these points and at the endpoints of the interval ([-1, 1]): [ f(-1) = (-1)^5 - (-1)^3 - 1 = -1 ] [ f(0) = 0^5 - 0^3 - 1 = -1 ] [ f(1) = 1^5 - 1^3 - 1 = -1 ] [ f\left(\sqrt{\frac{3}{5}}\right) = \left(\sqrt{\frac{3}{5}}\right)^5 - \left(\sqrt{\frac{3}{5}}\right)^3 - 1 ] [ f\left(\sqrt{\frac{3}{5}}\right) \approx -1.296 ] [ f\left(-\sqrt{\frac{3}{5}}\right) = \left(-\sqrt{\frac{3}{5}}\right)^5 - \left(-\sqrt{\frac{3}{5}}\right)^3 - 1 ] [ f\left(-\sqrt{\frac{3}{5}}\right) \approx -0.141 ]

Comparing these values, we find:

- Absolute maximum value: -0.141 (occurs at ( x = -\sqrt{\frac{3}{5}} ))
- Absolute minimum value: -1 (occurs at ( x = -1, 0, 1 ))

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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