Math Problem Statement

​c) Determine the​ interval(s) on which the function is decreasing. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

Solution

To determine the intervals on which a function is decreasing, you would typically:

  1. Find the derivative of the function, f(x)f'(x).
  2. Set the derivative equal to zero and solve for xx to find critical points, which may indicate changes in increasing or decreasing behavior.
  3. Analyze the sign of the derivative on intervals defined by the critical points:
    • If f(x)<0f'(x) < 0 on an interval, the function is decreasing on that interval.
    • If f(x)>0f'(x) > 0 on an interval, the function is increasing on that interval.

If you provide the specific function, I can calculate the derivative and determine the intervals where the function is decreasing.

Would you like further details on any of these steps? Here are some questions that may help clarify this topic:

  1. How do you find the derivative of a function?
  2. What are critical points, and how do they relate to a function’s increasing and decreasing intervals?
  3. How does the sign of a derivative affect the behavior of a function on an interval?
  4. Can a function have multiple decreasing intervals?
  5. How do you handle functions with no real critical points?

Tip: Remember, critical points occur where the derivative is zero or undefined.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivative
Critical Points
Intervals of Increase/Decrease

Formulas

f'(x) = 0
Sign of the derivative (f'(x) < 0 indicates decreasing function)

Theorems

First Derivative Test
Critical Point Theorem

Suitable Grade Level

Grades 11-12