Fermat’s theorem

Definition

Fermat's theorem states that if a function has a local maximum or minimum at some point, and the derivative exists at that point, then the derivative must be zero. It is essential for finding critical points in calculus.

5 Must Know Facts For Your Next Test

1. Fermat's theorem applies to differentiable functions.
2. If $f'(c)=0$, then $c$ is called a critical point.
3. Critical points are potential locations for local maxima and minima.
4. The theorem does not guarantee that a function with $f'(c)=0$ has a local extremum at $c$; it only indicates potential extremum points.
5. To determine if a critical point is an actual maximum or minimum, further tests like the second derivative test may be required.

Review Questions

• What does Fermat's theorem state about the derivative of a function at its local extrema?
• Why are critical points important in determining maxima and minima?
• What additional steps might be necessary after finding a critical point using Fermat's theorem?

Related terms

Critical Point: A point on the graph of a function where its derivative is zero or undefined.

Second Derivative Test: A method to determine whether a critical point is a local minimum, local maximum, or neither by analyzing the sign of the second derivative.

$f'(x)$: The first derivative of the function $f(x)$, representing the rate of change of $f(x)$ with respect to $x$.