Find the extreme values of the function and where they occur.

y =

Question

Find the extreme values of the function and where they occur.

y =

BlackTom AI · Accepted Answer

We are given y = 1/(x^2 + 1) and asked to find the extreme values and where they occur.
Option 1: Local maximum value is 1 at x = 0. This is plausible because the function is symmetric and achieves its highest value at x = 0. However, while it is a local maximum, the question typically emphasizes the absolute extreme values first; we should verify whether this is indeed a local maximum and whether it is the absolute maximum as well.
Option 2: Absolute maximum value is 1 at x = 0.5, absolute minimum value is -1 at x = 0.5. This is incorrect on multiple counts: the function y = 1/(x^2+1) is always positive, so it cannot have a minimum of -1. Also, the maximum 1 occurs at x = 0, not at x = 0.5. There is no negative value and the x = 0.5 claim for both max and min is wrong.
Option 3: Absolute maximum value is 1 at x = 0. This matches the behavior of the function: since x^2 ≥ 0, the denominator x^2+1 ≥ 1, with equality at x = 0, giving y = 1. For |x| large, denominator grows and y decreases toward 0, so 1 is the largest value the function attains, occurring at x = 0. This is correct as the absolute maximum.
Option 4: Absolute minimum value is 1.25 at x = 0.5. This is incorrect because y = 1/(x^2+1) is ≤ 1 for all x, and specifically at x = 0.5 the value is y = 1/(0.25+1) = 1/1.25 = 0.8, not 1.25, so the stated minimum and value are both wrong.
In summary, the only option that correctly identifies the absolute maximum value and its location is Option 3, since the function attains its largest value of 1 at x = 0 and decreases as |x| increases.

Find the extreme values of the function and where they occur. y =

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