Maxima and Minima Calculator

Enter a polynomial or mathematical function (e.g. x^3 - 3x^2 + 2) and specify the variable and a search interval to find its local maxima, local minima, and critical points. The Maxima and Minima Calculator evaluates your function's first derivative, identifies where f'(x) = 0, and returns the x-coordinates, y-values, and point classification for each extremum.

Leading coefficient of the highest-degree term

Constant term (used for cubic and quartic)

Used only for quartic functions

Left boundary of the search interval

Right boundary of the search interval

Results

Critical Points Found

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Local Maximum — x value

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Local Maximum — f(x) value

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Local Minimum — x value

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Local Minimum — f(x) value

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Global Maximum on Interval

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Global Minimum on Interval

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Function Curve with Critical Points

Results Table

Frequently Asked Questions

What are local maxima and minima?

A local maximum is a point where the function value is higher than all nearby points, and a local minimum is where it's lower than all nearby points. They are collectively called local extrema or extreme points. These differ from global extrema, which are the highest or lowest values over the entire domain or specified interval.

What is absolute (global) minima and maxima?

The absolute minimum is the smallest value a function takes over a given interval or its entire domain, while the absolute maximum is the largest. Every continuous function on a closed interval is guaranteed to have both an absolute maximum and minimum, according to the Extreme Value Theorem.

How do you determine local maxima and minima mathematically?

You find critical points by setting the first derivative f'(x) = 0 and solving for x. Then you apply the Second Derivative Test: if f''(x) > 0 at a critical point, it's a local minimum; if f''(x) < 0, it's a local maximum; if f''(x) = 0, the test is inconclusive and you must use the First Derivative Test to examine sign changes.

What is a critical point?

A critical point is any x-value where the first derivative f'(x) equals zero or is undefined. Critical points are candidates for local extrema, but not every critical point is an extremum — some are inflection points where the function changes concavity without a peak or valley.

Can a function have multiple local maxima and minima?

Yes. Higher-degree polynomials can have several local maxima and minima alternating across their domain. For example, a cubic polynomial can have at most one local maximum and one local minimum, while a quartic can have up to two local minima and one local maximum (or vice versa).

What is the difference between a local maximum and a global maximum?

A local maximum is only the highest point in its immediate neighborhood, while the global (absolute) maximum is the highest point over the entire interval or domain being considered. A function can have many local maxima, but only one global maximum per interval.

How does the Second Derivative Test work?

After finding a critical point x₀ where f'(x₀) = 0, compute f''(x₀). If f''(x₀) > 0, the curve is concave up at that point, confirming a local minimum. If f''(x₀) < 0, the curve is concave down, confirming a local maximum. If f''(x₀) = 0, the test is inconclusive and additional analysis is needed.

Why does this calculator use polynomial coefficients instead of a raw equation input?

Parsing arbitrary mathematical expressions in a browser-based tool requires a full symbolic math engine. By entering coefficients directly for standard polynomial forms (quadratic, cubic, quartic), the calculator can apply exact derivative formulas and deliver reliable results without ambiguity in expression parsing.

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