It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function on [a,b] and f(a) and f(b) have opposite signs, the intermediate-value theorem guarantees that there exists at least one solution of the equation f(x) = 0 in [a,b]. Explain in words why there exists exactly one solution in (a,b) if in addition, f is differentiable in (a,b), and f'(x) is eithe strictly positive or strictly negative throughout (a,b). Choose the correct answer below. O A. The graph of a monotonic function is a parabola that opens downward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b) such that f(c) = 0. O B. The graph of a monotonic function is a horizontal line. Given that the function is either strictly positive or strictly negative throughout (a,b) and that f(a) and f(b) have opposite signs, there exists at least one point ce(a,b) such that f(c) = 0. OC. The graph of a monotonic function is a parabola that opens upward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b) such that f(c) = 0. O D. The graph of a monotonic function is either decreasing or increasing. A horizontal line will intersect the graph at only one point. Given that the function is either strictly positive or strictly negative throughout (a,b) and that f(a) and f(b) have opposite signs, there exists cE(a,b) such that f(c) = 0.
It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function on [a,b] and f(a) and f(b) have opposite signs, the intermediate-value theorem guarantees that there exists at least one solution of the equation f(x) = 0 in [a,b]. Explain in words why there exists exactly one solution in (a,b) if in addition, f is differentiable in (a,b), and f'(x) is eithe strictly positive or strictly negative throughout (a,b). Choose the correct answer below. O A. The graph of a monotonic function is a parabola that opens downward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b) such that f(c) = 0. O B. The graph of a monotonic function is a horizontal line. Given that the function is either strictly positive or strictly negative throughout (a,b) and that f(a) and f(b) have opposite signs, there exists at least one point ce(a,b) such that f(c) = 0. OC. The graph of a monotonic function is a parabola that opens upward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b) such that f(c) = 0. O D. The graph of a monotonic function is either decreasing or increasing. A horizontal line will intersect the graph at only one point. Given that the function is either strictly positive or strictly negative throughout (a,b) and that f(a) and f(b) have opposite signs, there exists cE(a,b) such that f(c) = 0.
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...
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![It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function on [a,b] and f(a) and f(b) have opposite signs, the intermediate-value theorem guarantees
that there exists at least one solution of the equation f(x) = 0 in [a,b]. Explain in words why there exists exactly one solution in (a,b) if in addition, f is differentiable in (a,b), and f'(x) is either
strictly positive or strictly negative throughout (a,b).
Choose the correct answer below.
O A. The graph of a monotonic function is a parabola that opens downward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b)
such that f(c) = 0.
O B. The graph of a monotonic function is a horizontal line. Given that the function is either strictly positive or strictly negative throughout (a,b) and that f(a) and f(b) have opposite
signs, there exists at least one point cE(a,b) such that f(c) = 0.
OC. The graph of a monotonic function is a parabola that opens upward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b) such
that f(c) = 0.
O D. The graph of a monotonic function is either decreasing or increasing. A horizontal line will intersect the graph at only one point. Given that the function is either strictly positive or
strictly negative throughout (a,b) and that f(a) and f(b) have opposite signs, there exists cE(a,b) such that f(c) = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22a06b3b-3d0f-4964-a947-5f582a332765%2F743a53ff-e3a1-4a7b-901d-d4d9a7ebd726%2Fkjnlqjt_processed.png&w=3840&q=75)
Transcribed Image Text:It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function on [a,b] and f(a) and f(b) have opposite signs, the intermediate-value theorem guarantees
that there exists at least one solution of the equation f(x) = 0 in [a,b]. Explain in words why there exists exactly one solution in (a,b) if in addition, f is differentiable in (a,b), and f'(x) is either
strictly positive or strictly negative throughout (a,b).
Choose the correct answer below.
O A. The graph of a monotonic function is a parabola that opens downward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b)
such that f(c) = 0.
O B. The graph of a monotonic function is a horizontal line. Given that the function is either strictly positive or strictly negative throughout (a,b) and that f(a) and f(b) have opposite
signs, there exists at least one point cE(a,b) such that f(c) = 0.
OC. The graph of a monotonic function is a parabola that opens upward. A horizontal line will intersect the graph at two points. Therefore, there exists at least one point cE(a,b) such
that f(c) = 0.
O D. The graph of a monotonic function is either decreasing or increasing. A horizontal line will intersect the graph at only one point. Given that the function is either strictly positive or
strictly negative throughout (a,b) and that f(a) and f(b) have opposite signs, there exists cE(a,b) such that f(c) = 0.
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