Fermat's Theorem — Definition, Formula & Examples
Fermat's Theorem is the rule that if a function has a local high or low point at an interior point of its domain (not an endpoint), and the function has a derivative there, then that derivative must be zero. In other words, the tangent line at that point is horizontal.
Let be defined on an open interval containing . If has a local maximum or local minimum at , and exists, then . Consequently, every local extremum of at an interior point occurs at a critical number, meaning a point where or does not exist.
Key Formula
Where:
- = A function defined on an open interval containing c
- = An interior point where f has a local maximum or minimum
- = The derivative of f at c, assumed to exist
Worked Example
Problem: Use Fermat's Theorem to find every possible location of a local extremum of on the real line.
Step 1: Differentiate. The function is a polynomial, so it is differentiable everywhere. Every point of the real line is an interior point.
Step 2: By Fermat's Theorem, any local extremum must occur where . Solve that equation.
Step 3: Evaluate at each candidate. Fermat's Theorem only narrows the list. A test such as the second derivative test, with , confirms the type of each point.
Answer: The only possible local extrema are at and . Since and , has a local maximum of 16 at and a local minimum of at .
Why It Matters
Fermat's Theorem is the reason the Closed Interval Method in AP Calculus works. On a closed interval, you only need to check the endpoints and the critical numbers to find absolute extrema. It is also the key step in proving Rolle's Theorem, which in turn proves the Mean Value Theorem.
Common Mistakes
Mistake: Using the converse: assuming that if , then must have a local extremum at .
Correction: The theorem works in one direction only. For , , but is increasing through , so there is no extremum there. Treat points where as candidates, then confirm each one with the first or second derivative test.