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Absolute Extrema in Calculus

Learn about absolute maximum and minimum values of functions on closed intervals using Critical points and End points methods. Practical examples included.

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Absolute Extrema in Calculus

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  1. 14.1 – Absolute Extrema

  2. Absolute maximum and minimum A function f has an absolute (global) maximumat c if f (c) ≥ f (x) for all x in the domain. The number f (c) is called the maximum value of f. A function f has an absolute (global) minimum at c if f (c) ≤ f (x) for all x in the domain. The number f (c) is called the minimum value of f.

  3. The Extreme Value Theorem If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c) and absolute minimumvalue f (d) at some numbers c and d in [a, b] • An absolute extremum occurs at two places: • Critical points • End points

  4. Finding Absolute Extrema on a Closed Interval [a,b] 1. Find the critical points of f on (a, b). • Compute the value of f at each of • the critical points on (a,b) • the endpointsa and b. 3. The largest (smallest) of the values in Step 2 is the absolute maximum (minimum) value.

  5. Examples Locate the absolute extrema of the function on the closed interval f(x) = x3 – 12x on [0,4] g(x) = x ln(x+3) on [0,3]

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