Math Problem Statement

The graph of the derivative of a function f is shown. The graph has horizontal tangent lines at x = -1, x = 1, and x = 3. At which of the following values of x does f have a relative maximum?

Solution

Solution:

The problem involves analyzing the graph of the derivative f(x)f'(x) of a function f(x)f(x). We are asked to determine at which points the original function ff has a relative maximum.

Key Observations:

  1. Behavior of the derivative f(x)f'(x):

    • A relative maximum occurs at a point x=cx = c if:
      • f(c)=0f'(c) = 0 (i.e., a horizontal tangent in ff).
      • f(x)f'(x) changes from positive to negative at x=cx = c.

    This means we are looking for where the derivative crosses the x-axis (or touches it with a horizontal tangent) and the slope transitions from positive to negative.


Step-by-Step Analysis:

Looking at the graph of f(x)f'(x):

  • At x=2x = -2:

    • f(2)=0f'(-2) = 0 (it touches the x-axis).
    • Sign change: f(x)f'(x) changes from positive to negative.
    • Therefore, f(x)f(x) has a relative maximum at x=2x = -2.
  • At x=1x = 1:

    • f(1)=0f'(1) = 0 (horizontal tangent line).
    • Sign change: f(x)f'(x) changes from positive to negative.
    • Therefore, f(x)f(x) has a relative maximum at x=1x = 1.
  • At x=3x = 3:

    • f(3)=0f'(3) = 0 (horizontal tangent line).
    • Sign change: f(x)f'(x) changes from positive to negative.
    • Therefore, f(x)f(x) has a relative maximum at x=3x = 3.

Answer:

Since f(x)f(x) has relative maxima at x=2x = -2, x=1x = 1, and x=3x = 3, the correct answer is:

(E) 2-2, 1, and 3


Do you need further clarifications or details on this?
Here are five related questions you might find helpful:

  1. How do you determine whether a critical point is a maximum or minimum using the first derivative test?
  2. What does it mean when the derivative f(x)f'(x) touches but does not cross the x-axis?
  3. Can a function have a maximum or minimum at a point where f(x)f'(x) is undefined?
  4. What is the significance of the second derivative f(x)f''(x) in determining maxima and minima?
  5. How does the behavior of the derivative affect the concavity of the original function?

Tip: Always check where the derivative changes signs to determine relative maxima and minima—it provides a clearer picture than relying on values alone!

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivative
Relative Maximum
First Derivative Test

Formulas

f'(x) = 0 at critical points
f'(x) changes from positive to negative at a relative maximum

Theorems

First Derivative Test

Suitable Grade Level

Grades 11-12