Let f: R → R be a function that is differentiable on (a, ∞), where a is any realconstant. Let g: RR be a function defined byg(x) = f(x+1)-f(x).If limx→∞ f'(x) = 0, prove that limz+ g(x) = 0 by using the Mean Value Theo-rem.

Answered: Image /qna-images/answer/52e8869d-dd7e-4127-8415-b46414a61e29.jpg

Answered: Let f: R → R be a function that is…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

See similar textbooks

Similar questions

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
Let f: RR be a function that is differentiable on (a, o), where a is any real constant. Let g: RR be a function defined by g(x) = f(x + 1) - f(x). If limx→∞ f'(x) = 0, prove that limo g(x) = 0 by using the Mean Value Theo- rem.
17. Sketch the graph of f(x): = 2x + 2, 3 x³ = x, x ≤ -1 −1 and find the derivative. Determine whether f satisfies the conditions of the mean-value theorem on the interval [-3,2] and, if so, find the numbers c that satisfy the conclusion of the theorem.
Let f(x) = Vx +7 and g(x) = x +3 a) Find (2) (2) show work. b) Give the domain of the function (-) x) in interval notation
(5) Let f: R→R, and suppose there exist a positive integer M such that 0 < f(x) < M for all x ER. Use the limit definition of the derivative (either version) to prove that g(x) = x²f(x) + 2x is differentiable at x = 0. Do not use differentiation rules (such as power rule, product rule, ...).
2. a. f(x) = (x+2)/(x +4). Use the Mean Value Theorem on the interval [0, 4] to find all values of x in the interval (0, 4) such that the value of the first derivative of f at those points is (f(b) – f(a))/(b – a). b. For the interval [-5, 0], fb) - fa))(b — а)-GG0) — f-5)(0 — (-5)) = (½ – 3)/(0 + 5) = -½ but, since the first derivative of f(x) is always positive, nowhere on that interval does the first derivative of f(x) = -½. Thus the Mean Value Theorem does not hold for f(x) on this interval. Why?
Suppose f is defined on [a, b] and g is defined on [b, c] with f(b) = g(b). Then define f(z) if a
II. Consider the functions f(x)=- -9 **-1 2x and g(x)= (3x+2x-1 ; x=-1 ; x=-1 Define the new function 3/(x)- g(x), then determine is it continuous at x=-1?
Let P(x0, y0) be an arbitrary point on the graph of f that f′(x0) ≠ 0, as shown in the figure. Verify each statement.
Define F : R → R and G : R → Z by the following formulas: F(x) = (x^2)/3 and G(x) = ⌊x⌋ for every x ∈ R . (a) (G ∘ F)(2) = (b) (G ∘ F)(−3) = (c) (G ∘ F)(4) =
Consider the function f(x) = In (x² - 4) a.) Use the First Derivative Test or the Second Derivative Test to find any points that are local maxima and local minima. Include both x and y coordinates. b.) Find any inflection points of f. Make sure to include both x and y coordinates.
Let f be a function that is continuous on [1, 7] and differentiable on (1,7). Suppose 2 < f'(x) < 3 for all x in (1,7) and f(2) 3. What are the minimum and maximum possible values for f(4)?

SEE MORE QUESTIONS

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

Let f: R → R be a function that is differentiable on (a, ∞), where a is any real constant. Let g: R → R be a function defined by g(x) = f(x+1)-f(x). If limz+∞ f'(x) = 0, prove that limz→∞ g(x) = 0 by using the Mean Value Theo- rem.