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Absolute Minimum — Definition, Formula & Examples

Absolute Minimum, Absolute Min
Global Minimum, Golbal Min

The lowest point over the entire domain of a function or relation.

Note: The first derivative test and the second derivative test are common methods used to find minimum values of a function.

 

Graph of a curve showing an absolute minimum (lowest point) and two relative minimums labeled on the x-y plane.

 

See also

Relative minimum, local minimum, relative maximum, local maximum, absolute maximum, global maximum, extremum

Key Formula

f(c)≤f(x)for all x∈Df(c) \leq f(x) \quad \text{for all } x \in D
Where:
  • ff = The function being analyzed
  • cc = The input value where the absolute minimum occurs
  • f(c)f(c) = The absolute minimum value of the function
  • DD = The domain of the function

Worked Example

Problem: Find the absolute minimum of f(x)=x3−3x+2f(x) = x^3 - 3x + 2 on the closed interval [−2,3][-2, 3].
Step 1: Find the derivative and set it equal to zero to locate critical points.
f′(x)=3x2−3=0  ⟹  x2=1  ⟹  x=−1 or x=1f'(x) = 3x^2 - 3 = 0 \implies x^2 = 1 \implies x = -1 \text{ or } x = 1
Step 2: Both critical points x=−1x = -1 and x=1x = 1 lie within the interval [−2,3][-2, 3], so both are candidates.
Step 3: Evaluate f(x)f(x) at each critical point and at both endpoints of the interval.
f(−2)=(−2)3−3(−2)+2=−8+6+2=0f(-2) = (-2)^3 - 3(-2) + 2 = -8 + 6 + 2 = 0
Step 4: Evaluate at the critical points.
f(−1)=(−1)3−3(−1)+2=−1+3+2=4f(1)=(1)3−3(1)+2=1−3+2=0f(-1) = (-1)^3 - 3(-1) + 2 = -1 + 3 + 2 = 4\\[6pt]f(1) = (1)^3 - 3(1) + 2 = 1 - 3 + 2 = 0
Step 5: Evaluate at the right endpoint.
f(3)=(3)3−3(3)+2=27−9+2=20f(3) = (3)^3 - 3(3) + 2 = 27 - 9 + 2 = 20
Step 6: Compare all four values: f(−2)=0f(-2) = 0, f(−1)=4f(-1) = 4, f(1)=0f(1) = 0, f(3)=20f(3) = 20. The smallest value is 00.
Answer: The absolute minimum value is 00, occurring at both x=−2x = -2 and x=1x = 1.

Frequently Asked Questions

Can a function have more than one absolute minimum?
A function can have only one absolute minimum value, but that value can occur at more than one input. For example, f(x)=x2f(x) = x^2 on [−3,3][-3, 3] has an absolute minimum value of 00 at x=0x = 0 only, but f(x)=cos⁡(x)f(x) = \cos(x) on [0,4π][0, 4\pi] reaches its absolute minimum value of −1-1 at both x=πx = \pi and x=3πx = 3\pi.
Does every function have an absolute minimum?
Not always. A continuous function on a closed interval [a,b][a, b] is guaranteed to have an absolute minimum by the Extreme Value Theorem. However, functions on open intervals or unbounded domains may not. For instance, f(x)=1/xf(x) = 1/x on (0,1)(0, 1) has no absolute minimum because the function grows without bound as xx approaches 00 from the right—and it never actually reaches that bound.

Absolute Minimum vs. Relative (Local) Minimum

An absolute minimum is the lowest value of ff over the entire domain—no other output is smaller anywhere. A relative minimum is the lowest value in some small neighborhood around a point; the function may take even smaller values elsewhere. Every absolute minimum is also a relative minimum (unless it occurs at an endpoint), but a relative minimum is not necessarily the absolute minimum. For example, a function might dip to f=2f = 2 at one local minimum and dip to f=−1f = -1 at another; only f=−1f = -1 is the absolute minimum.

Why It Matters

Finding the absolute minimum is essential in optimization problems across science, engineering, and economics. When you want to minimize cost, energy, distance, or error, you need the absolute minimum—not just a local dip. The Extreme Value Theorem guarantees its existence on closed intervals, giving you a reliable strategy: check critical points and endpoints.

Common Mistakes

Mistake: Checking only the critical points and forgetting to evaluate the function at the endpoints of a closed interval.
Correction: On a closed interval [a,b][a, b], the absolute minimum can occur at an endpoint. Always evaluate f(a)f(a) and f(b)f(b) alongside all critical points, then compare.
Mistake: Confusing the absolute minimum value with the xx-value where it occurs.
Correction: The absolute minimum value is the output f(c)f(c), not the input cc. When asked 'what is the absolute minimum?', give the yy-value. When asked 'where does it occur?', give the xx-value.

Related Terms