Understanding Local Maximum | Calculus and Function Behavior Analysis

When f ‘(x) changes from positive to negative, f(x) has a

local maximum

local maximum.

In calculus, the derivative of a function f(x) at a certain point represents the instantaneous rate of change of the function at that point. When the derivative f ‘(x) changes from positive to negative, it means that the function is increasing up to a certain point and then starts decreasing.

At the point where f ‘(x) changes from positive to negative, f(x) reaches a critical point. This critical point can either be a local maximum or a local minimum. To determine which one it is, we can analyze the behavior of the function around that point.

If we observe that f(x) is increasing before the critical point and then starts decreasing after it, then f(x) has a local maximum at that point. This means that the value of f(x) at this point is greater than the values of f(x) on both sides of the critical point in a small interval.

It is important to note that while a local maximum represents a high point on the graph of the function, it may not necessarily be the highest point across the entire function. To identify the highest point on the graph of f(x) across its entire domain, we would need to examine the behavior of the function at the boundaries or consider the possibility of a global maximum.

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