Quadratic Function Maximum and Minimum
Find the maximum and minimum of ax²+bx+c on an entered closed interval. Compare both endpoints and the vertex, with a graph and candidate table.
YOUR CALCULATION
Result
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Calculation steps
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How to calculate
A continuous quadratic on a closed interval reaches its extrema at an endpoint or at its vertex when that vertex lies inside the interval. Evaluate those candidates and compare the values. A linear or constant function needs only endpoint checks.
Vertex xᵥ=−b/(2a); evaluate f(lower), f(upper), and f(xᵥ) when lower≤xᵥ≤upper.
Worked example
For f(x)=x²−4x+1 on [0,5], the vertex is (2,−3). The endpoints give f(0)=1 and f(5)=6. Therefore the minimum is −3 at x=2, the maximum is 6 at x=5 and the range is [−3,6].
Assumptions and input details
This is a bounded, closed-domain calculation, not an unrestricted or open-interval optimization. Bounds and coefficients must be finite and within ±1e6. Use a=0 for a linear or constant function; nonzero a must have magnitude at least 1e−9. Equal bounds give a one-point domain. A constant function reaches both extrema at every x in the interval. Results and graph coordinates use floating-point arithmetic and rounded display.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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