# What are the absolute extrema of # f(x)= x^5 -x^3+x^2-7x in [0,7]#?

Minimum:

Maximum:

We're asked to find the global minimum and maximum values for a function in a given range.

which happens to be the only critical point.

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To find the absolute extrema of ( f(x) = x^5 - x^3 + x^2 - 7x ) in the interval ([0, 7]), first, evaluate the function at the critical points and endpoints within the interval. Then, compare the function values to determine the maximum and minimum.

- Find the critical points by setting the derivative equal to zero and solving for ( x ).
- Evaluate ( f(x) ) at the critical points and endpoints within the interval.
- Identify the maximum and minimum values among these function values.

The critical points occur where the derivative is zero or undefined. Taking the derivative of ( f(x) ) yields ( f'(x) = 5x^4 - 3x^2 + 2x - 7 ). To find critical points, set ( f'(x) = 0 ) and solve for ( x ). Then evaluate ( f(x) ) at these points and the endpoints of the interval ([0, 7]).

Finally, determine which of these function values is the maximum and minimum within the interval.

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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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