Let f be a function that is continuous on some interval I ....

Let $f$ be a function that is continuous on some interval $I .$ Suppose $c$ is a critical number of $f$ and $(a, b)$ is some open interval in $I$ containing $c$. Prove that if $f^{\prime}(x)$ has the same sign on both sides of $c,$ then $f(c)$ is neither a local maximum value nor a local minimum value.

Key Concepts

Critical Number

A critical number of a function is a point in its domain where the derivative is either zero or fails to exist. These points are important because they are potential locations where the function's behavior might change, such as where a local maximum or minimum could occur.

Local Extremum

A local extremum (either maximum or minimum) is a point in the function's domain where the function attains a value that is greater than or less than all other values in some neighborhood around that point. Determining if a point is a local extremum involves examining the behavior of the function in an interval surrounding the point.

Continuity

Continuity of a function on an interval means that, roughly speaking, the function has no breaks, jumps, or holes in that interval. This property is vital when analyzing local behavior around points because many of the standard tests for extrema, including those involving derivatives, rely on the function being continuous.

Sign of the Derivative

The sign of the derivative indicates whether the function is increasing or decreasing at a particular interval. If the derivative is positive, the function is increasing; if it is negative, the function is decreasing. Consistency in the sign of the derivative on both sides of a critical point suggests that the function does not experience a change in increasing/decreasing behavior at that point, which is a key factor in determining whether the point is a local extremum.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical number is a local maximum, minimum, or neither. By examining how the sign of the derivative changes as you pass through the critical point, one can infer the presence or absence of a local extremum. In cases where the derivative does not change sign, the test indicates that the critical number is not associated with a local maximum or minimum.

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