Test- a 78.pdf

Test- a 78.pdf - Examples 4.31(i Consider the polynomial...

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Examples 4.31. (i) Consider the polynomial function f : R R defined by f(x)=x 4 8x 3 +22x 2 24x+7. Then f is differentiable, and one can easily check that f (x) = 4x 3 24x 2 +44x 24 = 4(x 1)(x 2)(x 3). Therefore, f (x) 0 if x 3 or 1 x 2, whereas f (x) 0 if x 1 or 2 x 3. Thus, f is monotonically increasing on [1, 2] and on [3, ), whereas f is

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monotonically decreasing on [2, 3] and on ( −∞ , 1]. In fact, since f vanishes only at x = 1, 2, and 3, we see that f is strictly increasing on (1,2) and on (3, ), whereas f is strictly decreasing on (2,3) and on ( −∞ ,1). Notice that in an
• Spring '14
• ThomasW.Appelquist
• Monotonic function, Convex function, polynomial function

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