How do you locate the absolute extrema of the function on the closed interval #y = 3 - abs(x - 3)#, on the interval [-1, 5]?

Answer 1

Find the first derivative and any critical values.
Find the y-value for each critical value and the y-value at the endpoints. The largest is abs. max, the smallest is abs min.

Note, the graph of any absolute value equation has a V shape, this one is upside down, shifted right 3 units and up 3 units. The corner of the V is at x = 3. #f(-1)=-1# the absolute min #f(3)=3# the absolute max #f(5)=1#
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Answer 2

To locate the absolute extrema of the function (y = 3 - |x - 3|) on the interval ([-1, 5]), follow these steps:

  1. Find the critical points by setting the derivative of the function equal to zero and solving for (x). However, since (y = 3 - |x - 3|) has corners at (x = 3) and (x = -1), there are only two critical points: (x = -1) and (x = 5).

  2. Evaluate the function at the critical points as well as at the endpoints of the interval ([-1, 5]). This will give you the (y) values corresponding to each critical point and endpoint.

  3. Compare the (y) values obtained in step 2. The highest (y) value corresponds to the absolute maximum, and the lowest (y) value corresponds to the absolute minimum.

Therefore, to locate the absolute extrema:

  • Evaluate the function at (x = -1), (x = 5), and (x = 3).
  • Compare the resulting (y) values to determine the absolute maximum and absolute minimum on the interval ([-1, 5]).
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Answer from HIX Tutor

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.

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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