Problem #129 Theoren!"suchTheorem 22. (The Mean Value Theorem) Suppose f' crists for creryE (a.b) andf is contimuous on a, b). Then there is a real number CE (a, b)of davege ratef(b) – f(a) = ÖVer Mtenad q.b.roveち-a f'lc) =%3Db- aTheorem 23. (Rolle's Theorem) Suppose f' crists for crery r E (a.b). fis continuous on a. b), andUse toprovemuTtof(a) = f(b).-veThen there is a real number c E (a, b) such thatf'(c) = 0.Problem 129. Show that if f'(x) <0 for every x in the interval (a, b) then fis decreasing on (a, b).Nested Interval Property of the Real Number System (NIP). Supposewe have two sequences of real numbers (xn) and (yn) satisfying the followingconditions:Chapte1. 21 < a2 < a3 s... [(*n) is non-decreasing]2. y1 2 y2 2 y3 2... [(yn) is non-increasing]3. V n, In SYn4. limn00 (Yn – an) = 0

Recall that, a function is said to be decreasing function on (a,b) if for each x1 and x2 in (a,b)…

Answered: Problem 129. Show that if f'(x) <0 for…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.3: The Addition And Subtraction Formulas

Problem 76E

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Conside the function f(x) = cubed root of x Does f(x) satisfy the hypothesis of mean value theorem on the interval [0,8]? Explain.
The graph of a function f is shown. The x y-coordinate plane is given. The curve begins at y = 1 on the positive y-axis, appears to go horizontally right, passes through the approximate point (3, 1), goes up and right becoming more steep, and ends at the point (5, 3) nearly vertical. Does f satisfy the hypotheses of the Mean Value Theorem on the interval [0, 5]? Yes, because f is continuous on the open interval (0, 5) and differentiable on the closed interval [0, 5].Yes, because f is increasing on closed interval [0, 5]. Yes, because f is continuous on the closed interval [0, 5] and differentiable on the open interval (0, 5).No, because f is not differentiable on the open interval (0, 5).No, because f does not have a minimum nor a maximum on the closed interval [0, 5].No, because f is not continuous on the open interval (0, 5). If so, find a value c that satisfies the conclusion of the Mean Value Theorem on that interval. (If an answer does not exist, enter DNE.) c = 3
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Problem 129. Show that if f'(x) <0 for every x in the interval (a, b) then f is decreasing on (a, b).