Math Problem Statement

Assume the function f is differentiable over the interval ​(minus−infinity∞​,infinity∞​). ​Thus, it is smooth and continuous for all real numbers x and has no corners or vertical tangents. Classify the following statement as either true or false. If you choose​ false, explain why. The function f can have a exactly one extreme value but no points of inflection. Question content area bottom Part 1 Select the correct choice below. True ​False, a critical value never occurs in a function without a point of inflection

Solution

The correct answer is True.

Here's the reasoning:

  • A function can have exactly one extreme value (either a maximum or a minimum) but no points of inflection.
  • The presence of an extreme value does not necessarily imply the existence of a point of inflection. A point of inflection occurs when the concavity of a function changes (from concave up to concave down or vice versa), which is determined by the second derivative f′′(x)f''(x).
  • However, a function can have an extreme value without having a change in concavity, and therefore, without a point of inflection. For example, a quadratic function like f(x)=−x2f(x) = -x^2 has a maximum at x=0x = 0, but no points of inflection because the concavity does not change (it is concave down everywhere).

Thus, the statement is true because a function can have one extreme value and no points of inflection.

Do you want further details or have any questions?

Here are 5 related questions that expand on this concept:

  1. What conditions must be met for a point to be classified as an inflection point?
  2. Can a function have more than one inflection point and still no extreme values?
  3. How do you determine whether a critical point is a maximum, minimum, or saddle point?
  4. Is it possible for a function to have inflection points but no extreme values?
  5. Can a function be concave but have no inflection points?

Tip: To identify points of inflection, look at where the second derivative changes sign.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Extreme Values
Inflection Points

Formulas

First derivative test: f'(x) = 0 at critical points
Second derivative test: f''(x) determines concavity and points of inflection

Theorems

Extreme Value Theorem
Concavity Test using second derivative

Suitable Grade Level

Grades 11-12

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