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Extrema

Extrema is the collective term for all maximum and minimum values of a function. A single maximum or minimum is called an extremum, and together they are called extrema.

The extrema of a function ff are the points in its domain where ff attains its greatest or least values, either globally (absolute extrema) or within a neighborhood of a point (local or relative extrema). An absolute maximum is the highest value of ff on its entire domain, while a local maximum is a point where f(c)≥f(x)f(c) \geq f(x) for all xx near cc. The same logic applies to minimum values. Extrema can occur at critical points (where f′(x)=0f'(x) = 0 or f′(x)f'(x) is undefined) or at the endpoints of a closed interval.

Key Formula

f′(c)=0orf′(c) is undefinedf'(c) = 0 \quad \text{or} \quad f'(c) \text{ is undefined}
Where:
  • f′(c)f'(c) = the derivative of $f$ evaluated at $c$
  • cc = a critical point where an extremum may occur

Worked Example

Problem: Find all extrema of f(x)=x3−3x+2f(x) = x^3 - 3x + 2 on the interval [−2,2][-2, 2].
Step 1: Find the derivative of f(x)f(x).
f′(x)=3x2−3f'(x) = 3x^2 - 3
Step 2: Set the derivative equal to zero and solve for the critical points.
3x2−3=0  ⟹  x2=1  ⟹  x=−1 or x=13x^2 - 3 = 0 \implies x^2 = 1 \implies x = -1 \text{ or } x = 1
Step 3: Evaluate f(x)f(x) at each critical point and at both endpoints of the interval.
f(-2) = -8 + 6 + 2 = 0$$ $$f(-1) = -1 + 3 + 2 = 4$$ $$f(1) = 1 - 3 + 2 = 0$$ $$f(2) = 8 - 6 + 2 = 4
Step 4: Identify the extrema by comparing all the values. The largest values are the maxima and the smallest values are the minima.
Answer: The absolute maximum value is 44, occurring at x=−1x = -1 and x=2x = 2. The absolute minimum value is 00, occurring at x=−2x = -2 and x=1x = 1. Additionally, x=−1x = -1 is a local maximum and x=1x = 1 is a local minimum.

Visualization

Why It Matters

Finding extrema is central to optimization problems — situations where you need to maximize profit, minimize cost, or determine the best dimensions for a design. In calculus, extrema also help you understand the overall shape and behavior of a function, which is essential for accurate curve sketching and analysis.

Common Mistakes

Mistake: Assuming that every critical point is an extremum.
Correction: A critical point where f′(x)=0f'(x) = 0 could be an inflection point instead. For example, f(x)=x3f(x) = x^3 has f′(0)=0f'(0) = 0, but x=0x = 0 is not an extremum. Use the first or second derivative test to verify.
Mistake: Forgetting to check the endpoints of a closed interval.
Correction: On a closed interval [a,b][a, b], absolute extrema can occur at the endpoints, not just at critical points. Always evaluate f(a)f(a) and f(b)f(b).

Related Terms