What is a Critical Point in Calculus? Graphing, Derivatives & Examples Explained

What is a Critical Point in Calculus? Graphing, Derivatives & Examples Explained
Master the concept of Critical Points in calculus with this quick, visual guide!
Whether you are studying AP Calculus AB/BC, College Calculus, or high school math, understanding critical numbers and how derivatives behave is key to finding local extrema, maximums, minimums, and analyzing graphs.
Key Takeaways
Definition of Critical Point: A number c in the domain of a function f(x) where the derivative f'(c) = 0 or f'(c) is undefined.
Horizontal Tangents: When f'(c) = 0, the graph has a flat, horizontal tangent line (often signaling a peak or valley).
Non-Differentiable Points: When f'(c) does not exist (like the sharp corner or cusp at x = 0 in f(x) = |x|), it is still classified as a critical point.
Timestamps
0:00 – What is a Critical Point?
0:01 – Where Derivative Equals Zero (f'(c) = 0)
0:04 – Where Derivative is Undefined (Absolute Value Example)
0:08 – Summary & Both Cases Explained
Why Are Critical Points Important?
Finding critical numbers is the first step in the First Derivative Test and Second Derivative Test. They help you identify:
Local Maximums and Minimums
Absolute Extrema on closed intervals
Points of interest on function graphs
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