Absolute Extrema in Calculus: Finding Maximum and Minimum Values, Exercises of Calculus

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Absolute Extrema

Michael Freeze MAT 151 UNC Wilmington Summer 2013

Section 6.1 :: Absolute Extrema

Absolute Maximum or Minimum

Let f be a function defined on some interval. Let c be a number in the interval. Then f (c) is the absolute minimum value of f on the interval if

f (x) ≥ f (c)

for every x in the interval.

Absolute Maximum or Minimum

A function has an absolute extremum at c if it has either an absolute maximum value or an absolute minimum value there.

Finding Absolute Extrema

To find absolute extrema for a function f continuous on a closed interval [a, b]: 1 Find all critical numbers for f in (a, b). 2 Evaluate f for all critical numbers in (a, b). 3 Evaluate f for the endpoints a and b of the interval [a, b]. 4 The largest value found in Step 2 or 3 is the absolute maximum value for f on [a, b], and the smallest value found is the absolute minimum value for f on [a, b].

Finding Absolute Extrema

Find the absolute extrema for the given continuous function on the indicated closed, bounded interval. f (x) = x^2 + 4x + 5, − 3 ≤ x ≤ 1

Finding Absolute Extrema

Find the absolute extrema for the given continuous function on the indicated closed, bounded interval. f (x) = x^3 + 3x^2 + 1, − 3 ≤ x ≤ 2

Finding Absolute Extrema

Find the absolute extrema for the given continuous function on the indicated closed, bounded interval. f (x) = x^3 + 3x^2 + 1, − 3 ≤ x ≤ 2

− 4 − 3 − 2 − 1 1 2 3

− 10

10

20

30

Finding Absolute Extrema

Find the absolute extrema for the given continuous function on the indicated closed, bounded interval. f (x) =

x^2 x − 1

, − 2 ≤ x ≤

− 3 − 2 − 1 1 2

− 2

− 1

1

Finding Absolute Extrema

Find the absolute extrema for the given continuous function on the indicated closed, bounded interval. f (x) =

x^2 − 9

, 0 ≤ x ≤ 2

Critical Point Theorem

Suppose a function f is continuous on an

interval I and that f has exactly one critical

number in the interval I, located at x = c.

If f has a relative maximum at x = c,

then this relative maximum is the

absolute maximum of f on the interval I.

If f has a relative minimum at x = c,

then this relative minimum is the

absolute minimum of f on the interval I.