2. Which of the following functions satisfy the hypothesis of Rolle's Theorem on the interval [0,2]? 1 (a) f(x) = |x-1| (b) f(x) = |x - 1|| (c) f(x) = x2 – 2x (d) f(x) = 2x + 3

Q: If we apply Rolle's Theorem to the function f(x) = 2x – 12x – 3 on the interval [1, 5], how many…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: The following statement is false, explain why. The function g(x) = x^2 on the interval [-1,1] has…

A:

Q: Verify that the three conditions of the hypothesis of Rolle's Theorem hold for the function f (x) =…

A: 1. f(x) is continuous on the closed interval [a,b] 2. f(x) is differentiable on the open interval…

Q: What is the value of k so that the function f(x) is continuous at x=2 O-3 O - 3/2 1/2 0 1 T-2k f(x)…

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Q: If fx) is a continuous function on [4,7] and F(x) is an anti-derivative of f(x) such that F(4)=9 and…

A: Given f(x) is continuous on [4, 7] and F(x) is the anti-derivative of f(x)                 F (4) = 9…

Q: 2. (a) Suppose that f is a differentiable function such that |f'(x)| < c for every x. Show that for…

A: To Prove: a. If f is a differentiable function such that f'(x)≤c for every x.Then for any real…

Q: Given the function f(x) = {2ª 2x x 2 which statement is FALSE at x=2? O The derivative from the…

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Q: let                    x^2+1   if x is less than 1 f(x)=                   x+1        if x is…

A: Consider the given function fx=x2+1  if x&lt;1x+1   if x≥1. The point is x = 1. Obtain the left hand…

Q: 16. Suppose that f is differentiable at c and that f'(c) + 0. Show that there exists a ð > 0 such…

A:

Q: 18. Suppose that f is a differentiable function such that f(g(x))= x, and f'(x)= |3D 1 1+(f(x))².…

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Q: 4 The function f (x) = x 2 2x Select one: a. continuous at x = 2 and differentiable at r = 2. O b.…

A: A function is continuous if Left hand limit is equal to right hand limit.

Q: 1. Test the continuity of the following functions at given points? x² -1 -1 <x<0 2x 0 <x<1 (a) fx)=…

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Q: If f(x) is a continuous function on [3,7] and F(x) is an anti-derivative of f(x) such that F(3)=7…

A: Given :f(x) is continuous function 3,7Fx is anti-derivative of fxF3=7 and F7=3To determine…

Q: 18. Suppose that f is a differentiable function such that f(g(x)) = x, and f'(x) = 1 1+(f(x))². Show…

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Q: 2. Which of the following functions satisfy the hypothesis of Rolle's Theorem on the interval [0,…

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Q: 1. Suppose f is a function which satisfies f(x + y) = f(x) + f (y) for all x,y e R. What is the…

A: Let f is a function which satisfies f(x+y) = f(x) + f(y)  for all x,y∈R   f(x+y) = f(x) + f(y) ,   ∀…

Q: Let f(x) be a differentiable function for -3<x<1. Over this interval f(x) has exactly one critical…

A: for maxima or minima, we need to check second derivative of the function.

Q: ) If we apply Rolle's Theorem to the function f(x)=2x^2−8x−4 on the interval [0,4], how many values…

A: Given: fx=2x2-8x-4 on the interval 0,4 To find the value of C using Rolle's Theorem.

Q: 35. Given that f(z) is a continuous even function, then A 0 B D

A:

Q: 1 For a function f(x), we have f(x) = 1+ x? and f(0) = 0 prove that x +y f(x) +f(y) =f 1-xy

A: We will prove as following.

Q: 3. Find the points of relative extrema of the following functions on the specified domain: (a)…

A: Introduction: Evaluation of maximum and minimum value of a function is one of the most important…

Q: 7. Suppose that f: R R is differentiable at c and that f(c) = 0. Show that g(x):= |f(x)| is…

A:

Q: 2. Find the critical point in the domain of following functions 2 (a) f(x) = v9 – x² (b) f(x) = x +…

A: As per company guideline we are suppose to answer only one question with three subpart at a time .…

Q: 9. Show the function f(z) = 1 + 2i+2Re(z) is differentiable or not differentiable at any points.

A: Solution

Q: Suppose that a function h satisfies −x^2 ≤ h(x) ≤ x^2 for all x. Prove that h is differentiable at x…

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Q: For what value of the constant c is the function f continuous on the interval (-0,00). f(x)={ cx+1 ,…

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Q: 1. Employ the following methods to find the maximum of f(x) = 4x – 1.8x² + 1.2x – 0.3x* (a)…

A: Given f(x)=4x-1.8x2+1.2x3-0.3x4. To find: (a) find the maximum using golden search method.…

Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: 1 If f(x) is a non-negative continuous function for all x, such that f(x) + + f(x+1/2) = 1,0 ≤ x ≤ -…

A: Given that fx is a non-negative continuous function for all x, such that fx+fx+12 = 1 for 0 ≤ x ≤…

Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

Q: -1.5 0 1.5 For what value of constant c is the function k(x) continuous at x = 0 if k(x)= No such…

A:

Q: 2.4 Use the definition of the derivative of complex function to show that f(z) is not dif-…

A: The Complex Derivative Definition: Let f be a function defined on a neighborhood of z0 ∈ ℂ. If…

Q: The critical points of the function f(x) = 9x² – 2xª occur at %3D O x = 0, x = ±0.75 O z = 0, x =…

A: Given: fx=9x2-2x4

Q: 10.C. If f is defined and continuous on R to R and if f(xo) > 0, then is f positive on some…

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Q: 1. Is a function f(x)=√x differentiable at a) X = 1 b) X = 0 c) X = -1

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: 6- if f(x) is a continuous function at x = 0, then a = 1- cos 4x x0 V16+ Vx -4 3.

A:

Q: If we apply Rolle's Theorem to the function f(x) = 2x² - 12x - 4 on the interval [1, 5], how many…

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Q: if x1 O No, since g(x) is undefined at x=1. O Yes, and g'(1)=3. O Yes, and g'(1)=8. O No, since g(x)…

A: Given

Q: 2. Consider the function f with domain (-2, 0): 3|22 – 1| if – 2<r<1 2.r2 + 2x – 4 f(x) = 3 if r = 1…

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Q: (8) Suppose that f(x) and g(x) are differentiable and one-to-one and f(2) = –1, f'(2) = -3, f(-1) =…

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Q: Prove that the function f(x) = cos x is strictly decreasing on the interval [0,1]. %3!

A: 5) Given that, fx=cosx

Q: Explain why the following function satisfies the hypotheses of Rolle's Theorem on the given…

A: As the function is a polynomial function so it is differentiable on (-2,-1) and continuous on…

Q: 3 - x, x<2 Q.No.15: Is f(x) = {+2 x22 continuous at every x € R. x+1 xs1 -.

A: Continuity of a function.

Q: 1. Consider the function z = f(x, y). If fa(a, b) and fy(a, b) both exist, and they are equal, then…

A: 1. Given: z=fx,y and suppose that fxa,b and fya,b both exists and they are equal and it is…

Q: 2. For which of the following functions and domains does the Extreme Value Theorem guarantee the…

A: The Extreme value theorem states that if a function is continuous on a closed interval [a,b], then…

Q: 7. Find all values of x on the graph of f(x)=(x-1)´(2x+1) at which the tangent line is horizontal.…

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Q: 1 if z 0. Is F(x) is continuous at I = 0? Justify your answer.

A: Given that for x&lt;=0, F(x) = (x - 1)/4 Then as x tends to zero from the negative side of zero, we…

Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.

Q: (i) Let ƒ be the function defined on (-o, 0) by f(x) = r$(3 – x)}. (2 – x) -2 Given that f'(x) = and…

A: