Prove Fermat's Theorem for the case in which f has a local minimum at c.

Name: Problem 71, Chapter 3: Calculus
Uploaded: 2018-03-15T05:50:11-08:00
Duration: 2 min 40 s
Channel: Wen Zheng
Description: Prove Fermat's Theorem for the case in which f has a local minimum at c.

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Prove Fermat's Theorem for the case in which f has a local...

Key Concepts

Differentiability

Differentiability at a point implies that the function has a well-defined tangent at that point and that the derivative, which represents the instantaneous rate of change, exists. The existence of the derivative is crucial in Fermat's Theorem because it allows the application of the limit definition of the derivative to infer properties about the function at its local extremum.

One-Sided Derivatives

One-sided derivatives consider the limit of the difference quotient as the variable approaches the point from only one direction (either from the left or the right). In the proof of Fermat's Theorem, analyzing the one-sided limits at a point where the function has a local minimum shows that the right-hand derivative is nonnegative and the left-hand derivative is nonpositive; when these are equal (as required by differentiability), both must equal zero.

Fermat's Theorem

Fermat's Theorem states that if a function has a local extremum (either maximum or minimum) at a point and is differentiable at that point, then its derivative at that point must be zero. This theorem is a fundamental result in calculus that helps identify potential points where a function changes behavior, commonly known as critical points.

Local Minimum

A local minimum is a point in the domain of a function where the function's value is less than or equal to the values at nearby points. This condition implies that within some neighborhood around the point, every value of the function is at least as large as the function's value at the minimum, establishing a scenario for using derivative reasoning.

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