How do you find the absolute maximum and minimum on a closed interval?

Use the Closed Interval Method: (1) Find f'(x) and set it to 0 to find critical points, (2) Evaluate f(x) at each critical point within [a,b] and at both endpoints, (3) The largest value is the absolute maximum, the smallest is the absolute minimum.

What is the difference between local and absolute extrema?

Local extrema are the highest or lowest points in a small neighborhood. Absolute extrema are the overall highest or lowest points on the entire interval. An absolute extremum is always a local extremum (or an endpoint), but not vice versa.

Can absolute extrema occur at endpoints?

Yes. By the Extreme Value Theorem, a continuous function on [a,b] attains its absolute max and min. These can occur at critical points or at the endpoints a and b. Always check endpoints.

How do you find critical points of a polynomial?

Take the derivative f'(x), then solve f'(x) = 0. For a quadratic derivative, use the quadratic formula. For a linear derivative (from a quadratic f), just solve the linear equation. Critical points are where the function could change from increasing to decreasing.

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Extrema Calculator

Find the absolute maximum and minimum of a function on a closed interval. Step-by-step solutions using the Closed Interval Method.

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f(x) (polynomial, e.g. x^3 - 3x + 1)

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How It Works

What are absolute extrema?

The absolute maximum is the highest point and the absolute minimum is the lowest point of a function on a closed interval [a, b]. Every continuous function on a closed interval has both.

The Closed Interval Method

Find f'(x), set it equal to zero to get critical points. Evaluate f at each critical point and at both endpoints. The largest value is the absolute max; the smallest is the absolute min.

Critical points

A critical point is where f'(x) = 0 or f'(x) is undefined. For polynomials, critical points are where the derivative equals zero. These are the candidates for local max/min.

Local vs absolute extrema

A local max/min is the highest/lowest point in a neighborhood. An absolute max/min is the highest/lowest point on the entire interval. Absolute extrema can occur at endpoints or critical points.

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