Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

c. At what points, if any, does f assume local maximum or minimum values?

f′(x) = x(x − 1)

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To find the points where the function f assumes local maximum or minimum values, we need to identify the critical points. Critical points occur where the derivative f'(x) is zero or undefined. Start by setting the derivative equal to zero: f'(x) = x(x - 1) = 0.

Solve the equation x(x - 1) = 0 to find the critical points. This equation can be factored into two parts: x = 0 and x - 1 = 0. Solving these gives the critical points x = 0 and x = 1.

Next, determine whether each critical point is a local maximum, minimum, or neither by using the first derivative test. This involves analyzing the sign of f'(x) around each critical point.

Choose test points in the intervals determined by the critical points: for example, pick a point less than 0, between 0 and 1, and greater than 1. Evaluate the sign of f'(x) at these test points to determine the behavior of f(x).

If f'(x) changes from positive to negative at a critical point, f has a local maximum there. If f'(x) changes from negative to positive, f has a local minimum. If there is no sign change, the critical point is neither a maximum nor a minimum.

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주요 개념

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local maxima or minima. To find them, set the derivative equal to zero and solve for x. In this case, solve f′(x) = x(x − 1) = 0 to find the critical points.

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, one can infer the behavior of the function. If f′ changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.

Local Maximum and Minimum

A function has a local maximum at a point if the function value at that point is greater than at nearby points. Conversely, a local minimum occurs if the function value is less than at nearby points. Identifying these points involves analyzing the critical points and using tests like the First Derivative Test to confirm the nature of each point.

Analyzing Functions from DerivativesAnswer the following questions about the functions whose derivatives are given in Exercises 1–14:c. At what points, if any, does f assume local maximum or minimum values?f′(x) = x(x − 1)