Suppose f(x) = x° + 3x + 1. In this problem, we will show that f has exactly one root (or zero) in the interval [-2, -1]. (a) First, we show that f has a root in the interval (-2, –1). Since f is a continuous function on the interval [-2, -1] and f(-2) =| 59 and f(-1) =|-1 the graph of y = f(x) must cross the -axis at some point in the interval by the intermediate value theorem . Thus, f has at least one root in the interval [-2, -1]. (b) Second, we show that f cannot have more than one root in the interval [-2, –1] by a thought b in the interval [-2, –1] with experiment. Suppose that there were two roots x = a and x = a < b. Then f(a) = f(b) =| Since f is continuous differentiable on the interval (-2, –1), by Rolle's theorem + on the interval [-2, –1] and there would exist a point c in interval (a, b) so that f'(c) = 0. However, the only solution to f'(x) = 0 is x = %3D which is not in the interval (a, b), since (a, b) C[-2, –1]. Thus, f cannot have more than one root in [-2, –1].

Suppose f(x) = x° + 3x + 1. In this problem, we will show that f has exactly one root (or zero) in the interval [-2, -1]. (a) First, we show that f has a root in the interval (-2, –1). Since f is a continuous function on the interval [-2, -1] and f(-2) =| 59 and f(-1) =|-1 the graph of y = f(x) must cross the -axis at some point in the interval by the intermediate value theorem . Thus, f has at least one root in the interval [-2, -1]. (b) Second, we show that f cannot have more than one root in the interval [-2, –1] by a thought b in the interval [-2, –1] with experiment. Suppose that there were two roots x = a and x = a < b. Then f(a) = f(b) =| Since f is continuous differentiable on the interval (-2, –1), by Rolle's theorem + on the interval [-2, –1] and there would exist a point c in interval (a, b) so that f'(c) = 0. However, the only solution to f'(x) = 0 is x = %3D which is not in the interval (a, b), since (a, b) C[-2, –1]. Thus, f cannot have more than one root in [-2, –1].

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

See similar textbooks