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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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.
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.
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.
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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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.
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.
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.
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.