Relative Minimum — Definition, Graph & Examples

Q: What is the difference between a relative minimum and an absolute minimum?

A relative minimum is the lowest point in its immediate neighborhood — the function dips down and then rises back up around that point. An absolute minimum is the single lowest function value over the entire domain. Every absolute minimum is also a relative minimum (unless it occurs at an endpoint), but a relative minimum is not necessarily the absolute minimum because the function may reach even lower values elsewhere.

Q: Can a function have more than one relative minimum?

Yes. A function can have multiple relative minima. For example, a polynomial of degree 4 or higher can have several valleys, each producing its own relative minimum. Each one is simply the lowest point in its local neighborhood.

Q: When do you use the first derivative test vs. the second derivative test to find a relative minimum?

The second derivative test is often quicker: if f'(c) = 0 and f''(c) > 0, you immediately know c gives a relative minimum. However, if f''(c) = 0 the test is inconclusive, and you must fall back on the first derivative test, which checks whether f' changes sign from negative to positive around c. The first derivative test always works at critical points, while the second derivative test can sometimes fail.

Relative Minimum, Relative Min
Local Minimum, Local Min

The lowest point in a particular section of a graph.

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

Graph of y=f(x) showing a curve with one global minimum (far left, lowest) and two local minima labeled along the x-axis.

Key Formula

f'(c) = 0 \quad \text{and} \quad f''(c) > 0 \implies f(c) \text{ is a relative minimum}

Where:

$f(x)$ = The original function
$c$ = The x-value where the relative minimum occurs (a critical point)
$f'(c)$ = The first derivative evaluated at c; equals zero at a critical point
$f''(c)$ = The second derivative evaluated at c; if positive, the curve is concave up, confirming a relative minimum

Worked Example

Problem: Find the relative minimum of f(x) = x² − 6x + 10.

Step 1: Find the first derivative of f(x).

f'(x) = 2x - 6

Step 2: Set the first derivative equal to zero and solve for x to find the critical point.

2x - 6 = 0 \implies x = 3

Step 3: Find the second derivative and evaluate it at x = 3.

f''(x) = 2 \implies f''(3) = 2 > 0

Step 4: Since the second derivative is positive, the function is concave up at x = 3, confirming a relative minimum. Compute the function value.

f(3) = (3)^2 - 6(3) + 10 = 9 - 18 + 10 = 1

Answer: The relative minimum is the point (3, 1).

Another Example

This example uses a cubic function with two critical points and applies the first derivative test instead of the second derivative test to identify which critical point is a relative minimum and which is a relative maximum.

Problem: Find all relative minima of f(x) = x³ − 3x² − 9x + 5 using the first derivative test.

Step 1: Find the first derivative.

f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x - 3)(x + 1)

Step 2: Set f'(x) = 0 to find critical points.

3(x - 3)(x + 1) = 0 \implies x = 3 \text{ or } x = -1

Step 3: Apply the first derivative test at x = 3. Choose test points on either side: at x = 2, f'(2) = 3(2−3)(2+1) = 3(−1)(3) = −9 < 0. At x = 4, f'(4) = 3(4−3)(4+1) = 3(1)(5) = 15 > 0. The sign changes from negative to positive, so x = 3 is a relative minimum.

f'(2) = -9 < 0, \quad f'(4) = 15 > 0

Step 4: Check x = −1. At x = −2, f'(−2) = 3(−5)(−1) = 15 > 0. At x = 0, f'(0) = 3(−3)(1) = −9 < 0. The sign changes from positive to negative, so x = −1 is a relative maximum, not a minimum.

f'(-2) = 15 > 0, \quad f'(0) = -9 < 0

Step 5: Compute the function value at the relative minimum.

f(3) = 27 - 27 - 27 + 5 = -22

Answer: The only relative minimum is at the point (3, −22).

Frequently Asked Questions

What is the difference between a relative minimum and an absolute minimum?

A relative minimum is the lowest point in its immediate neighborhood — the function dips down and then rises back up around that point. An absolute minimum is the single lowest function value over the entire domain. Every absolute minimum is also a relative minimum (unless it occurs at an endpoint), but a relative minimum is not necessarily the absolute minimum because the function may reach even lower values elsewhere.

Can a function have more than one relative minimum?

Yes. A function can have multiple relative minima. For example, a polynomial of degree 4 or higher can have several valleys, each producing its own relative minimum. Each one is simply the lowest point in its local neighborhood.

When do you use the first derivative test vs. the second derivative test to find a relative minimum?

The second derivative test is often quicker: if f'(c) = 0 and f''(c) > 0, you immediately know c gives a relative minimum. However, if f''(c) = 0 the test is inconclusive, and you must fall back on the first derivative test, which checks whether f' changes sign from negative to positive around c. The first derivative test always works at critical points, while the second derivative test can sometimes fail.

Relative Minimum vs. Relative Maximum

	Relative Minimum	Relative Maximum
Definition	Lowest point in a local neighborhood of the graph	Highest point in a local neighborhood of the graph
First Derivative Test	f' changes from negative to positive	f' changes from positive to negative
Second Derivative Test	f'(c) = 0 and f''(c) > 0	f'(c) = 0 and f''(c) < 0
Graph shape at point	Concave up (opens upward like a cup)	Concave down (opens downward like a cap)
Also called	Local minimum	Local maximum

Why It Matters

Finding relative minima is essential in calculus courses, especially when you sketch curves or solve optimization problems. In applied contexts, a relative minimum can represent the lowest cost, shortest distance, or least amount of material needed in a particular scenario. AP Calculus AB/BC exams regularly ask you to identify and justify relative extrema using derivative tests.

Common Mistakes

Mistake: Assuming that every point where f'(x) = 0 is a relative minimum.

Correction: A zero derivative only identifies a critical point. You must verify with the first or second derivative test. The critical point could be a relative maximum or an inflection point instead.

Mistake: Confusing relative minimum with absolute minimum.

Correction: A relative minimum is only the lowest value in a nearby region. The function could have lower values elsewhere. On a closed interval, always compare relative minima with the endpoint values to determine the absolute minimum.

Related Terms

Absolute Minimum — The lowest value over the entire domain
Relative Maximum — The highest point in a local neighborhood
Absolute Maximum — The highest value over the entire domain
Extremum — General term for any maximum or minimum
First Derivative Test — Sign-change method to classify critical points
Second Derivative Test — Concavity method to classify critical points
Function — The object whose minima are being found
Graph of an Equation or Inequality — Visual representation showing minima as valleys