Optimization Calculator
Find the vertex (maximum or minimum) of a quadratic function f(x) = ax² + bx + c. Use this when solving real-world optimization problems like maximizing profit or minimizing cost.
About this calculator
A quadratic function f(x) = ax² + bx + c has exactly one vertex, which is either a global maximum (when a < 0) or a global minimum (when a > 0). The x-coordinate of the vertex is found using the vertex formula: x = -b / (2a). Substituting this back into the original function gives the optimal value: f(-b/2a) = a(-b/2a)² + b(-b/2a) + c. This result comes from completing the square or setting the first derivative equal to zero. When a > 0 the parabola opens upward and the vertex is a minimum; when a < 0 it opens downward and the vertex is a maximum. This principle underlies countless applied problems in economics, physics, and engineering.
How to use
Suppose a company's profit is modeled by f(x) = -2x² + 8x + 3, where x is units sold. Enter a = -2, b = 8, c = 3. The optimal x = -8 / (2 × -2) = -8 / -4 = 2. Now evaluate: f(2) = -2(2²) + 8(2) + 3 = -8 + 16 + 3 = 11. The maximum profit is 11, achieved at x = 2. Because a = -2 < 0, this vertex is a maximum, confirming you should sell 2 units for peak profit.
Frequently asked questions
How do you find the maximum or minimum of a quadratic function?
The maximum or minimum of f(x) = ax² + bx + c occurs at the vertex. First compute the x-coordinate with x = -b / (2a). Then substitute that value back into the function to get the optimal output. If a > 0 the result is a minimum; if a < 0 it is a maximum.
What is the difference between a maximum and minimum in quadratic optimization?
Whether the vertex is a maximum or minimum depends entirely on the sign of the leading coefficient a. When a > 0, the parabola opens upward and the vertex is the lowest point, a global minimum. When a < 0, the parabola opens downward and the vertex is the highest point, a global maximum. There is no other local extremum for a quadratic.
When should you use quadratic optimization in real-world problems?
Quadratic optimization applies whenever a relationship between two quantities follows a parabolic curve. Common examples include maximizing revenue (price × quantity where demand is linear), minimizing material cost for a fixed volume, and finding the peak height of a projectile. Any time a model has the form ax² + bx + c and you need the best possible value, the vertex formula gives the answer directly without calculus.