Lecture Notes 21: Local Extrema, Fermat’s
Theorem and Critical Points
Instructor: Anatoliy Swishchuk
Department of Mathematics & Statistics
University of Calgary, Calgary, AB, Canada
MATH 265 ’University Calculus I’
L01 Winter 2017
Outline of Lecture
1. Local Extrema
2. Fermat’s Theorem
3. Critical Points
4. Examples
Local Extrema
A real-valued function f has a local maximum at x0 if f (x0) is
the largest value of f near x0; in other words, f (x0) ≥ f (x) when
x is near x0.
A real-valued function f has a local minimum at x0 if f (x0) is
the smallest value of f near x0; in other words, f (x0) ≤ f (x)
when x is near x0.
A local extrema is either a local minimum or a local maximum.
Fermat’s Theorem
If f (x) has a local extremum at x = a and f is differentiable at
a, then f 0(a) = 0.
Thus, the only points at which a function can have a local maximum or minimum are points at which the derivative is zero (e.g.,
x2 at x = 0) or the derivative is undefined, or, does not exist
(e.g., |x| at x = 0).
However, when f 0(a) = 0, f does not necessarily have a maximum
or minimum at a. (E.g., x3 at x = 0 is zero, but x3 has no
maximum or minimum at x = 0).
Critical Points
Any value of x in the domain of f for which f 0(x) is zero or
undefined is called a critical point for f.
If f has a local maximum or minimum at a, then a is a critical
point for f.
Examples
Example 1 (Local minimum and maximum). Function f (x) =
√
3
x − x has local minimum at x = 3/3 and local maximum at
√
x = − 3/3.
Example 2 (Fermat’s Theorem). Function f (x) = cos x has local
maximum value of 1 infinite many times at x = 2nπ and local
minimum value of −1 infinite many times at x = (2n + 1)π,
therefore, f 0(2nπ) = sin(2nπ) = 0 and f 0((2n + 1)π) = sin((2n +
1)π) = 0.
Example 3 (Critical Points). Function f (x) = x3/5(4 − x) has
the following critical points: 3/2 and 0.
References
1) Calculus: Early Transcendental, 2016, An Open Text, by
David Guichard: https : //lalg1.lyryx.com/textbooks/CALCU LU S
1/ucalgary/winter2016/math265/Guichard
− Calculus − EarlyT rans − U of Calgary − M AT H265 − W 16.pdf
2) Optional Textbook: Essential Calculus, Early Transcendental,
2013, by J. Stewart, 2nd edition, Brooks/Cole