If f:[0, 3] → R is continuous, must its range be a closed interval? [HINT: Think about a constant function]

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: True/False: If ƒ is entire and |f(z)| > 2 for all z E C with |2| > 1, then f is a constant function.

A: If f is entire and fz≥2 for all z∈ℂ with z≥1Let gz=1fzas fz≠0 for any z∈ℂ⇒g is entire gz=1fz<12…

Q: Suppose that f(x) is differentiable function such that f(1)=3 and f(1)=4. If g(x)= f(x) -, then what…

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Q: Say if the following statement is true or false: If a function f is twice differentiable at the…

A: If c is a critical point of the function, then  f'c=0. To check whether the critical point…

Q: Find the absolute maximum and minimum values of the function over the indicated interval, and…

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Q: Assume that h(x) = f(g(x)) , where both f and g are differentiable functions. If g(-1)=2, g'(-1)=3,…

A: Given : hx = fgxg-1= 2g'-1 = 3f'2 = -4

Q: 2. ASSUME f(x) IS CONTINUOUS AND DIFFERENTIABLE ON AN OPEN INTERVAL AS GRAPHED. WHICH OF THESE…

A: Since you have posted multiple question in this question so I have solved first question as per our…

Q: Let y=f(X) defines a twice-differentiable function and let y=T(x) be the line tangent to the graph…

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Q: Suppose that f(x) is a differentiable function such that f'(4)=5 and f(4)=4. If g(x)=xf(x), then…

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Q: If f(x) is continuous in the interval [-2, 1], ƒ(−2) = −4, and f(1) = -1, can we use the…

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Q: Given a function f(x) that is continuous and differentiable, and a point a in its domain, we…

A: This question can be solve by taking approximation as x tends to a (more close)

Q: Use the limit definition for the derivative to calculate f'(x) for each of the following if f'(x)…

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Q: Which is the correct conclusion about the function h(x) ={V4- x2 if x - 3 if h has a removable…

A: Given function is: hx=-x-2    ;  if x<-24-x2     ;  if -2≤x<2x-3  ;  if x>2 We have to…

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: Let f be a differentiable function such that f(4) = 3 and f'(x) = -3x + 3. What is the approximation…

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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: Suppose that  ff"' is continuous and f'(c)  =f"(c) =0 , but f"'(c) >0. Does f have a local…

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Q: Suppose that f is a function such that f' (c) = 1 for some number c in the interval [a, b]. Is it…

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Q: 2. If F and G are differentiable functions such that F'(x) = G'(x), which of the following is true…

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Q: Suppose f : [a, b] → R is is continuous and such that fg = 0 for all continuous functions g : [a, b]…

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Q: 19. Suppose f is a piecewise-defined function, equal to g(x) if x 2, where g and h are contin- uous…

A: Consider the given piecewise function. f=gx      x<2hx      x≥2 The functions g(x) and h(x) are…

Q: Given a set E c (0,1) of measure 0, construct a continuous non-decreasing function f : [0, 1] → R…

A: We will use knowledge of measure theory and real analysis to answer this question. As instructed in…

Q: , for x 1 Let f(x) be a function de fined by f(x)3= Discuss the contuniuty of f andf".

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Q: (11) (C) For the function f(x) = x° – x, determine whether the hypothesis of Rolle's theorem hold on…

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Q: . Do you agree with the contention that the function f(x)= x +2 and g(x) = x2 -4/x-2  are the same…

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Q: Prove that if f(x) is a differentiable function everywhere and f'(x) # 1 for any value of x, then…

A: The objective here is to prove that the mentioned function f(x) has maximum one fixed point. Given:…

Q: Suppose f and g are differentiable functions on R such that,

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Q: 5. Consider the function g(x)=1 a) For what values of x is g(x) continuous? b) For what values of x…

A: In the question we have to find where function is continuous and differentiable.

Q: Sketch the graph of a continuous function y = f(x) that satisfies the following three conditions:…

A: The continuous finction y=f(x) has properties,f'(x)>0 for -2<x<2f'(x)<0 for x<-2 and…

Q: If we apply Rolle's Theorem to the function f(x) = 2x- 16x 6 on the interval [2, 6|, how many values…

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Q: - We define the function f: R -> R as follows. Let f(x) = x-2 for X2, and f(x) = a for x=2. What…

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Q: Suppose that f(x) is differentiable function such that f' (1)=8 and f (1)=4. If    g(x) = (f(x)) /…

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Q: 1. Why is f not a function from R to R if a) f(x)= 1/x? b) f(x) = T? c) f(x) =±V(x² + 1)?

A: We'll answer the first question since the exact one was not specified. Please submit a new question…

Q: Prove that if f(x)=c (a constant function), then f'(x)= 0

A: Given query is to prove that derivative of constant function is zero.

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: 5. Suppose that function f(x) satisfies f'(x) 0 at interval [a, b], then the graph of y = f(x) ( )…

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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: 4. Use the limit definition of the derivative to test whether the following function f(x), given by…

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Q: Any function f defined on R should have a relative extremum.   True or False

A: In the question we have to find whether the statement is true or not.

Q: If f is a function satisfying |f (x) – f(y)| < (x – y)? - for all x and y, prove that f is constant.

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Q: The piecewise function f(x) is defined below. f(x) = {kx² - 4, Is-2 x+ 2, xS-2 x>-2 For what value…

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Q: If we apply Rolle's Theorem to the function f(x) = 2x² - 8x - 1 on the interval [0, 4], how many…

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Q: 15. If f+g is differentiable at a, are f and g necessarily differentiable at a ? If f g and f are…

A: Let f be any function that is not differentiable  at the point a and define as: gx=-fx for all real…

Q: Consider the function f with domain (-2, 0): 3|a2 – 1| if – 2 < x < 1 2x2 + 2x – 4 f(x) = 3 if x = 1…

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Q: / Let f: R → R be a differentiable function with the property that f(-x) = f(x) for all x E R. Prove…

A: Let f:ℝ→ℝ be a differentiable function with the property that f-x=fx for all x∈ℝ.   To Prove:…

Q: Draw the graph of a continuous function f(x) where f(-2) = 0, f(2) = 0, but there is no value c€…

A: Given: f(-2)=0 ;     f(2)=0but there exists at least one c∈-2, 2…

Q: If f has an absolute maximum value over a closed interval [a, b], a, b e R at x = xo, where xo E (a,…

A: Topic:- function

Q: ⎪ ⎨ ⎪ ⎧x^2 − 1 , − 1 ≤ x < 0 2x , 0 < x < 1 1 , x = 1 −2x + 4 , 1 < x < 2 0 , 2 < x < 3 a) Is f…

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Q: On what domain is the function f(x) = 5+ V2x + 6 continuous? %3D

A: Let's find.