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An optimization calculator (or optimization problem calculator) helps you solve optimization problems and find maximum and minimum values. Use this free constrained optimization calculator and calculus optimization calculator to optimize functions with constraints. This optimization solver calculator supports optimisation calculator (British spelling) and optimize calculator functionality. Also known as applied optimization calculator or online optimization calculator.
Last updated: February 2, 2026
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Enter function to optimize like -x² + 4x + 5
Enter any constraints on the problem
Critical Points:
x = 2
Maximum:
f(2) = 9
Minimum:
None (parabola opens downward)
Second Derivative Test:
f''(x) = -2 < 0 (concave down, confirms maximum)
Analysis:
Quadratic function with negative leading coefficient has a maximum
Solution:
Maximum value occurs at vertex x = 2
Optimization Steps:
Methods supported
First & Second Derivative Tests
Uses derivatives to find critical points and determine extrema
Extrema types
Local & Absolute Extrema
Identifies all local maxima and minima plus absolute extrema
Critical point test
f'(x) = 0
Solves f'(x) = 0 to find candidates for optimization
Constraint methods
Substitution & Lagrange
Handles equality constraints using substitution or multipliers
Area problems
A = f(x) subject to constraint
Optimizes area with fixed perimeter or other constraints
Volume problems
Boxes, Cylinders, Cones
Optimizes volume for various geometric shapes
For function f(x) = -x² + 4x + 5 (optimization problem):
Critical Point
x = 2
Maximum Value
f(2) = 9
An optimization problem calculator (or optimization problems calculator) solves optimization problems by finding maximum and minimum values of functions. It uses calculus methods: finding critical points by setting f'(x) = 0, applying the second derivative test to classify extrema, and checking boundary conditions. An optimization solver calculator (or optimize calculator) automatically solves these problems using calculus techniques. This optimization problem calculator handles single-variable and constrained optimization problems.
A constrained optimization calculator (or optimization calculator with constraints) optimizes functions subject to constraints or restrictions. It uses methods like substitution (eliminating variables using constraints) or Lagrange multipliers. A calculus optimization calculator (or optimization calculus calculator) uses calculus methods to find maximum and minimum values, applying the first and second derivative tests. This constrained optimization calculator handles problems where variables must satisfy certain conditions.
An applied optimization calculator solves real-world optimization problems such as maximizing area with fixed perimeter, minimizing cost subject to constraints, or finding optimal dimensions. An optimisation calculator (British spelling of optimization) performs the same functions: finding maximum and minimum values using calculus methods. This applied optimization calculator translates word problems into mathematical functions and determines optimal solutions. Also known as online optimization calculator.
Our optimization calculator uses calculus derivative methods to find maximum and minimum values of functions. The calculator applies the first and second derivative tests to identify critical points and determine whether they represent maxima, minima, or saddle points.
Step 1: Find derivative f'(x)Step 2: Solve f'(x) = 0 for critical pointsStep 3: Calculate second derivative f''(x)Step 4: Test: f''(x) > 0 → minimum, f''(x) < 0 → maximumStep 5: Check endpoints if domain is restrictedThis systematic approach ensures accurate identification of all extrema. The second derivative test provides conclusive evidence for the nature of each critical point through concavity analysis.
Shows critical points, maxima, and minima on a function graph
Optimization is a fundamental application of differential calculus. Fermat's theorem states that if a function has a local extremum at an interior point, then the derivative at that point must be zero or undefined. The second derivative test uses concavity to distinguish between maxima and minima.
Need help with other calculus problems? Check out our derivative calculator and concavity calculator.
Get Custom Calculator for Your PlatformResult: Maximum value is f(2) = 9 at x = 2
Since f''(2) < 0, the parabola is concave down, confirming a maximum.
Maximize area A = xy with perimeter constraint 2x + 2y = 100
Maximum area: 625 sq units (square: 25 × 25)
Maximize volume V = x(12-2x)² for open box from 12" square
Maximum volume: 128 cu in at x = 2 inches
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