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In mathematics, a local maximum of a function is a point where the function reaches a peak in its neighborhood, which means the function value is greater than or equal to nearby values. For a function to have a local maximum at a particular point, the function value at that point should be greater than or equal to the function values for other points close to it.

This occurs around critical points, where the first derivative of the function either equals zero or is not defined. Consider the function given in the problem:

• \( f(x) = -|x^2 - 4| \).

At \( x = 2 \) and \( x = -2 \), the function behaves such that just before these points, the value stays the same as at the point, and just after, the value decreases. Thus, these points exhibit a local maximum characteristic.

A local maximum doesn't mean it is the highest point globally over the entire function, but rather it simply is the highest point over a small interval.

The absolute value function, denoted as \( |x| \), returns the non-negative value of a given real number \( x \). This implies that if \( x \) is negative, the absolute value function will multiply it by -1 to make it positive. This is important as it affects how functions that include absolute values behave.

For instance, in the function \( f(x) = -|x^2 - 4| \):

• If \( x^2 - 4 \geq 0 \), then \( |x^2 - 4| = x^2 - 4 \), resulting in \( f(x) = -(x^2 - 4) \).
• If \( x^2 - 4 < 0 \), then \( |x^2 - 4| = -(x^2 - 4) \), leading to \( f(x) = x^2 - 4 \).

This piecewise nature makes absolute value functions interesting as they can easily switch expressions across certain points, changing the function's behavior rapidly.

This abrupt change contributes to non-differentiability at certain critical points in the overall function.

Critical points of a function are crucial for analyzing its graphs and finding maxima and minima. They occur where the derivative of a function is zero or undefined. These points mark potential locations for local maxima, minima, or inflection points.

In the exercise function \( f(x) = -|x^2 - 4| \), the points \( x = 2 \) and \( x = -2 \) are critical because:

• The derivative at these points doesn't exist due to the abrupt behavior change in the absolute value function.
• Checking the behavior slightly before and after \( x = 2 \) shows a switch from an increasing to decreasing slope. This aligns \( x = 2 \) as a local maximum.

Discontinuities in derivatives, especially at such critical points, indicate the locations where the function isn't smooth and differentiable. Correspondingly, at critical points like \( x = 2 \) and \( x = -2 \), one can observe local maxima and nonsmooth behavior. Understanding how these points work helps in sketching accurate graphs and understanding function behavior.