Mathwords logoMathwords

Maximum of a Function

Maximum of a Function:

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

 

 

See also

Extremum

Key Formula

f′(c)=0andf′′(c)<0  ⟹  f(c) is a local maximumf'(c) = 0 \quad \text{and} \quad f''(c) < 0 \implies f(c) \text{ is a local maximum}
Where:
  • f(c)f(c) = The value of the function at the critical point c
  • f′(c)f'(c) = The first derivative of f at c, equal to zero at a critical point
  • f′′(c)f''(c) = The second derivative of f at c; if negative, the curve is concave down, confirming a local maximum

Worked Example

Problem: Find all maximum values of the function f(x) = -x² + 4x - 1 on the interval [0, 5].
Step 1: Take the first derivative and set it equal to zero to find critical points.
f′(x)=−2x+4=0  ⟹  x=2f'(x) = -2x + 4 = 0 \implies x = 2
Step 2: Use the second derivative test to classify the critical point.
f′′(x)=−2<0for all xf''(x) = -2 < 0 \quad \text{for all } x
Step 3: Since f''(2) < 0, the function is concave down at x = 2, so x = 2 is a local maximum. Evaluate f at this point.
f(2)=−(2)2+4(2)−1=−4+8−1=3f(2) = -(2)^2 + 4(2) - 1 = -4 + 8 - 1 = 3
Step 4: To find the absolute maximum on [0, 5], also evaluate f at the endpoints of the interval.
f(0)=−1,f(5)=−25+20−1=−6f(0) = -1, \quad f(5) = -25 + 20 - 1 = -6
Step 5: Compare all values: f(0) = -1, f(2) = 3, f(5) = -6. The largest value determines the absolute maximum.
max⁡{−1, 3, −6}=3\max\{-1,\, 3,\, -6\} = 3
Answer: The function has a local maximum and an absolute maximum at x = 2, where f(2) = 3.

Another Example

Problem: Find the maximum of f(x) = x³ - 3x on the interval [-2, 2].
Step 1: Find the critical points by setting the first derivative 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: Apply the second derivative test to each critical point.
f′′(x)=6x  ⟹  f′′(−1)=−6<0,f′′(1)=6>0f''(x) = 6x \implies f''(-1) = -6 < 0, \quad f''(1) = 6 > 0
Step 3: Since f''(-1) < 0, x = -1 is a local maximum. Since f''(1) > 0, x = 1 is a local minimum (not a maximum).
f(−1)=(−1)3−3(−1)=−1+3=2f(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2
Step 4: Evaluate f at the endpoints to determine the absolute maximum.
f(−2)=−8+6=−2,f(2)=8−6=2f(-2) = -8 + 6 = -2, \quad f(2) = 8 - 6 = 2
Step 5: Compare all candidate values: f(-2) = -2, f(-1) = 2, f(1) = -2, f(2) = 2. The absolute maximum value of 2 is achieved at both x = -1 and x = 2.
max⁡{−2, 2, −2, 2}=2\max\{-2,\, 2,\, -2,\, 2\} = 2
Answer: The local maximum is at x = -1 with f(-1) = 2. The absolute maximum value on [-2, 2] is also 2, occurring at both x = -1 and x = 2.

Frequently Asked Questions

What is the difference between a local maximum and an absolute maximum?
A local (relative) maximum is a point where the function value is greater than or equal to all nearby values — it's the highest point in its immediate neighborhood. An absolute (global) maximum is the single largest value the function achieves over its entire domain or a specified interval. Every absolute maximum is also a local maximum, but a local maximum is not necessarily the absolute maximum.
Can a function have more than one maximum?
A function can have many local maxima, but on a closed interval it has exactly one absolute maximum value (though that value may be achieved at more than one point). On an open interval or over all real numbers, an absolute maximum may not exist at all — for example, f(x) = -x⁴ + x² has local maxima but approaches negative infinity, while f(x) = x has no maximum.

Maximum of a Function vs. Minimum of a Function

A maximum is a point where the function value is highest (locally or globally), while a minimum is a point where the function value is lowest. Both are found using the same techniques: setting the first derivative to zero and applying the second derivative test. If f′′(c)<0f''(c) < 0, you have a local maximum; if f′′(c)>0f''(c) > 0, you have a local minimum. Together, maxima and minima are called extrema.

Why It Matters

Finding the maximum of a function is central to optimization — determining the best possible outcome under given constraints. Engineers maximize structural strength, businesses maximize profit, and scientists maximize efficiency using these techniques. Understanding how local and absolute maxima differ also prevents errors in real-world decisions where a "good" solution may not be the best overall solution.

Common Mistakes

Mistake: Assuming that every point where f'(x) = 0 is a maximum.
Correction: A zero derivative indicates a critical point, which could be a maximum, a minimum, or a saddle/inflection point. You must use the second derivative test or the first derivative test to determine which type it is.
Mistake: Forgetting to check the endpoints of a closed interval when finding the absolute maximum.
Correction: On a closed interval [a, b], the absolute maximum can occur at an endpoint even if no critical point is there. Always evaluate the function at every critical point and at both endpoints, then compare.

Related Terms