Determine Whether the Statement is True or False If F is Continuous at a So is f

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Found in: Page 247

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Rolle's Theorem is a special case of the

Mean Value Theorem .

(b) True or False: The Mean Value Theorem is so named

because it concerns the average (or "mean") rate of

change of a function on an interval.

(c) True or False: If f is differentiable on R and has an extremum

at x = −2, then f '(−2) = 0.

(d) True or False: If f has a critical point at x = 1, then

f has a local minimum or maximum at x = 1.

(e) True or False: If f is any function with f (2) = 0 and

f (8) = 0, then there is some c in the interval (2, 8)

such that f '(c) = 0.

(f) True or False: If f is continuous and differentiable on

[−2, 2] with f (−2) = 4 and f (2) = 0, then there is

some c ∈ (−2, 2) with f '(c) = −1.

(g) True or False: If f is continuous and differentiable on

[0, 10] with f '(5) = 0, then f has a local maximum or

minimum at x = 5.

(h) True or False: If f is continuous and differentiable on

[0, 10] with f '(5) = 0, then there are some values a

and b in (0, 10) for which f (a) = 0 and f (b) = 0.

(a) True

(b) True

(c)True

(d) True

(e) True

(f) False

(g) True

(h) True

See the step by step solution

Step by Step Solution

Step 1. (a) (a) True or False: Rolle's Theorem is a special case of theMean Value Theorem.

True; Rolle's theorem, in analysis, special case of the mean-value theorem of differential calculus. Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b .

Step 2. (b) True or False: The Mean Value Theorem is so named because it concerns the average (or "mean") rate of change of a function on an interval.

True; In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. The word Mean describe the average between two points that's why it is called Mean value theorem .

Step 3. (c) True or False: If f is differentiable on R and has an extremumat x = −2, then f '(−2) = 0.

True; If x = c is in the domain of f called critical point of f if f'(c) = 0 so the given statement is true .

Step 4. (d) True or False: If f has a critical point at x = 1, then f has a local minimum or maximum at x = 1.

True;

(a) f has a local maximum at x = c if there exists some δ > 0 such that f (c) ≥ f (x) for all x ∈ (c − δ, c + δ).

(b) f has a local minimum at x = c if there exists some δ > 0 such that f (c) ≤ f (x) for all x ∈ (c − δ, c + δ).

Step 5. (e) True or False: If f is any function with f (2) = 0 andf (8) = 0, then there is some c in the interval (2, 8)such that f '(c) = 0.

True ; According to the Rolle's theorem the given statement is true .

If f is any function with f (2) = 0 and f (8) = 0, then there is some c in the interval (2, 8) such that f '(c) = 0.

Step 6. (f) True or False: If f is continuous and differentiable on [−2, 2] with f (−2) = 4 and f (2) = 0, then there is some c ∈ (−2, 2) with f '(c) = −1.

False ; In this statement f'(c) must be zero according to the Rolle's theorem If f is continuous on [a, b] and differentiable on (a, b), and if f (a) = f (b) = 0, then there exists at least one value c ∈ (a, b) forwhich f '(c) = 0.

Step 7. (g) True or False: If f is continuous and differentiable on[0, 10] with f '(5) = 0, then f has a local maximum orminimum at x = 5.

True ; Given statement satisfied the conditions of Rolle's theorem .

If in function we put the value then it satisfied the conditions of Rolle's theorem so the given statement is true .

Step 8. (h) True or False: If f is continuous and differentiable on [0, 10] with f '(5) = 0, then there are some values a and b in (0, 10) for which f (a) = 0 and f (b) = 0

True ; Given statement satisfied the conditions of Mean value theorem so it is true .

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