1. Consider the function f: R→ R defined by f(x) = x³ − 3x² + 4x + 5 (a) Use intermediate value theorem to show that f has at least one root. (b) Use Rolle's theorem to show that f has at most one root. (So, by (a) and (b), f has exactly one root.) 2. Give an example ƒ : [a, b] → R such that f is differentiable on (a, b), and f(a) = f(b) but there exist no c f'(c) = 0.

1. Consider the function f: R→ R defined by f(x) = x³ − 3x² + 4x + 5 (a) Use intermediate value theorem to show that f has at least one root. (b) Use Rolle's theorem to show that f has at most one root. (So, by (a) and (b), f has exactly one root.) 2. Give an example ƒ : [a, b] → R such that f is differentiable on (a, b), and f(a) = f(b) but there exist no c f'(c) = 0.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 51E

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