Identifying Relative or Local Maximums of Functions: A Guide to Finding Critical Points and Analyzing Function Behavior

Relative or Local maximum

at which it changes from increasing to decreasing

A relative or local maximum refers to a point on a graph of a function where the function reaches the highest value within a certain interval but is not necessarily the highest value of the entire function.

To identify a relative or local maximum, we need to check the values of the function at points surrounding the point in question. First, we need to locate the critical points of the function by finding where the derivative of the function equals zero or does not exist. Then, we need to examine the behavior of the function on either side of the critical points to determine whether it increases or decreases.

If the function increases on both sides of the critical point, then the critical point represents a relative or local minimum. If the function decreases on both sides of the critical point, then the critical point represents a relative or local maximum. However, if the function increases on one side and decreases on the other side of the critical point, then the critical point is not a relative or local extremum but rather a point of inflection.

For example, consider the function f(x) = x^3 – 3x^2 – 9x + 10. To find the critical points, we need to differentiate the function and set it equal to zero:

f'(x) = 3x^2 – 6x – 9
0 = 3(x^2 – 2x – 3)
0 = 3(x – 3)(x + 1)

The critical points are x = 3 and x = -1. To determine whether these critical points are relative or local extrema, we need to examine the sign of the derivative on either side of each critical point:

For x < -1, f'(x) < 0, so the function is decreasing. For -1 < x < 3, f'(x) > 0, so the function is increasing.
For x > 3, f'(x) > 0, so the function is increasing.

Therefore, at x = -1, the function has a relative or local maximum, and at x = 3, the function has a relative or local minimum.

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