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Minimum of a Function — Definition, Formula & Examples

Minimum of a Function

Either a relative (local) minimum or an absolute (global) minimum.

 

 

See also

Extremum

Key Formula

f′(c)=0andf′′(c)>0  ⟹  f(c) is a local minimumf'(c) = 0 \quad \text{and} \quad f''(c) > 0 \implies f(c) \text{ is a local minimum}
Where:
  • f(x)f(x) = The function being analyzed
  • cc = The x-value where the minimum occurs
  • f′(c)f'(c) = First derivative of f at c (equals zero at a critical point)
  • f′′(c)f''(c) = Second derivative of f at c (positive indicates concave up, confirming a local minimum)

Worked Example

Problem: Find the minimum of the function f(x) = x² − 6x + 11.
Step 1: Take the first derivative and set it equal to zero to find critical points.
f′(x)=2x−6=0  ⟹  x=3f'(x) = 2x - 6 = 0 \implies x = 3
Step 2: Use the second derivative test to confirm this critical point is a minimum.
f′′(x)=2>0(concave up, so x=3 is a local minimum)f''(x) = 2 > 0 \quad \text{(concave up, so } x = 3 \text{ is a local minimum)}
Step 3: Evaluate the function at x = 3 to find the minimum value.
f(3)=(3)2−6(3)+11=9−18+11=2f(3) = (3)^2 - 6(3) + 11 = 9 - 18 + 11 = 2
Step 4: Since f(x) = x² − 6x + 11 is an upward-opening parabola, this local minimum is also the absolute minimum of the function over all real numbers.
Absolute minimum value=2 at x=3\text{Absolute minimum value} = 2 \text{ at } x = 3
Answer: The minimum of f(x) = x² − 6x + 11 is 2, occurring at x = 3. This is both the local and absolute minimum.

Another Example

Problem: Find all minimum values of f(x) = x³ − 3x on the closed interval [−3, 3].
Step 1: Find the critical points by setting the first derivative equal to zero.
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: Use the second derivative to classify each critical point.
f′′(x)=6x  ⟹  f′′(−1)=−6<0 (local max),f′′(1)=6>0 (local min)f''(x) = 6x \implies f''(-1) = -6 < 0 \text{ (local max)}, \quad f''(1) = 6 > 0 \text{ (local min)}
Step 3: Evaluate f at the critical points and the endpoints of the interval to find the absolute minimum.
f(−3)=−27+9=−18,f(−1)=2,f(1)=−2,f(3)=27−9=18f(-3) = -27 + 9 = -18, \quad f(-1) = 2, \quad f(1) = -2, \quad f(3) = 27 - 9 = 18
Step 4: Compare all values. The local minimum is f(1) = −2, but the absolute minimum on the interval is f(−3) = −18, which occurs at an endpoint.
Local minimum: f(1)=−2Absolute minimum: f(−3)=−18\text{Local minimum: } f(1) = -2 \qquad \text{Absolute minimum: } f(-3) = -18
Answer: There is a local minimum of −2 at x = 1, and the absolute minimum on [−3, 3] is −18 at the endpoint x = −3.

Frequently Asked Questions

What is the difference between a local minimum and an absolute minimum?
A local (relative) minimum is a point where the function value is smaller than at all nearby points. An absolute (global) minimum is the single smallest value the function achieves over its entire domain. Every absolute minimum is also a local minimum, but a local minimum is not necessarily the absolute minimum — the function may dip even lower somewhere else.
Can a function have more than one minimum?
A function can have multiple local minima. However, if an absolute minimum exists, its value is unique (though it can be achieved at more than one x-value). Some functions, like f(x) = x³, have no minimum at all because they decrease without bound.

Minimum of a Function vs. Maximum of a Function

A minimum is the lowest value a function reaches; a maximum is the highest value. Both are found using the same techniques — setting the first derivative to zero and checking critical points — but the second derivative test gives opposite signs: positive (concave up) for a minimum and negative (concave down) for a maximum. Together, minima and maxima are called extrema.

Why It Matters

Finding the minimum of a function is central to optimization, one of the most practical applications of calculus. Engineers minimize cost, material usage, or energy consumption by locating function minima. In physics, systems naturally settle into states of minimum potential energy, so this concept connects directly to how the physical world behaves.

Common Mistakes

Mistake: Assuming that every point where f'(x) = 0 is a minimum.
Correction: A zero derivative only identifies a critical point. You must use the second derivative test (or the first derivative test) to determine whether the critical point is a minimum, a maximum, or neither (such as an inflection point like x = 0 for f(x) = x³).
Mistake: Forgetting to check endpoints when finding the absolute minimum on a closed interval.
Correction: On a closed interval [a, b], the absolute minimum can occur at an endpoint, not just at a critical point. Always evaluate the function at both endpoints and at every critical point inside the interval, then compare all values.

Related Terms