Critical Points and Extrema

Finding Maxima and Minima

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Critical points are the x-values where a function can change from increasing to decreasing, decreasing to increasing, or behave in a special flat or sharp way. They are important because they help identify local maximum and minimum values, which appear in optimization, graph analysis, and real-world modeling. In calculus, critical points connect the shape of a graph to the derivative, giving a precise way to locate interesting behavior.

A critical point occurs where $f'(x) = 0$ or where $f'(x)$ does not exist, as long as $f(x)$ itself is defined there. After finding critical points, students test the sign of $f'(x)$ on nearby intervals to decide whether each point is a local maximum, local minimum, or neither. Some critical points are flat points where the graph levels out but does not turn around, so not every critical point is an extremum. This idea is the basis of the first derivative test and supports many applications in physics, economics, and engineering.

Key Facts

A critical point occurs at $x = c$ if $f(c)$ is defined and either $f'(c) = 0$ or $f'(c)$ does not exist.
If $f'(x)$ changes from positive to negative at $x = c$ , then $f(c)$ is a local maximum.
If $f'(x)$ changes from negative to positive at $x = c$ , then $f(c)$ is a local minimum.
If $f'(x)$ does not change sign at $x = c$ , then the critical point is not a local extremum.
To find absolute extrema on $[a, b]$ , compare $f(a)$ , $f(b)$ , and all critical point values inside $(a, b)$ .
Second derivative test: if $f'(c) = 0$ and $f''(c) > 0$ , local minimum; if $f''(c) < 0$ , local maximum.

Vocabulary

Critical point: A point on the graph where the derivative is zero or undefined, provided the function itself is defined there.
Local maximum: A point where the function value is greater than nearby function values.
Local minimum: A point where the function value is less than nearby function values.
Derivative: The derivative measures the slope of the function and shows how the function is changing at each point.
Increasing interval: An interval where the function values rise as $x$ increases, usually where $f'(x) > 0$ .

Common Mistakes to Avoid

Treating every point where $f'(x) = 0$ as a maximum or minimum, which is wrong because some critical points are flat points where the graph keeps increasing or keeps decreasing.
Ignoring points where the derivative is undefined, which is wrong because corners, cusps, and vertical tangents can still be critical points if the function exists there.
Using only the equation $f'(x) = 0$ without checking sign changes, which is wrong because extrema depend on how the derivative behaves on both sides of the point.
Forgetting to test endpoints when finding absolute extrema on a closed interval, which is wrong because the largest or smallest value can occur at an endpoint instead of a critical point.

Practice Questions

1 Find all critical points of $f(x) = x^3 - 3x^2 + 2$ and classify each as a local maximum, local minimum, or neither.
2 A function has derivative $f'(x) = (x - 1)(x + 3)$ . Find the critical points and determine the intervals where the function is increasing and decreasing.
3 A graph has a critical point at $x = 2$ where $f'(2) = 0$ , but $f'(x)$ is positive on both sides of $x = 2$ . Explain why this point is not a local extremum.