Maxima and minima of functions Lesson 2.2

Definitions Global extrema: If f(c) ≥ f(x) for all x in the domain of f, f(c) is the global maximum value of f. If f(c)≤ f(x) for all x in the domain of f, f(c) is the global minimum value of f. Local extrema: If f(c) ≥ f(x) for all x in some open interval containing c, f(c) is a local maximum value of f. If f(c)≤ f(x) for all x in some open interval containing c, f(c) is a local minimum value of f.

Example 1: Y = button Approximate the global and local maximum and minimum on each given domain for the function k defined by k(x) = -2x4 + 3x3 + 4x2 – 5x + 5 Set of all real numbers: -1 ≤ x ≤ 1 x < -2

k(x) = -2x4 + 3x3 + 4x2 – 5x + 5 Set of all real numbers: -no global min, local min. at x≈ .477, k(x) ≈3.747 - local/global max. at x ≈-.865, k(x) ≈9.257, - local max. at x ≈ 1.513, k(x) ≈ 6.502

k(x) = -2x4 + 3x3 + 4x2 – 5x + 5 -1 ≤ x ≤ 1 -local/global min at x≈ .477, k(x) ≈3.747 -local min at endpoint x=-1, k(x) = 9 -local/global max. at x ≈-.865, k(x) ≈9.257, - local max at endpoint, x = 1, k(x) = 5

k(x) = -2x4 + 3x3 + 4x2 – 5x + 5 x < -2 - there is no minimum since the function decreases without bound on the interval (-∞, -2). -There is no maximum because k(x) increases as x increases and there is no greatest value of x on this interval.

Find the extrema of f(t)=2t4 +4t + 1 Over [0,∞) - local/global min: t=0, f(t) = 1 - No local/global max. Over (-3,1): -local/global min: t≈-.787, f(t) ≈ -1.381 -No local/global max.

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