Let f(x) = { 3, 1< x < 2 0, 2 < a <5 and Let F(x) = { 3x, 1

Q: 1. Supposef is a continuous function on x 2 0,f'exists for x> 0, f(0) = 0 and f'is monotonically…

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Q: B) Test the functions continuity at the given poihts: (x²-x-6 1) f(x) 3D x * 3 ,at x 3, x 5 X-3 x =…

A: f(x)=x2-x-6x-3x≠35x=3

Q: Prove that the function given by f(x) =x3-3x2+3x-100 is increasing in R

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Q: snip

A: The given statement is true. Because the local maximum is the highest point on the curve. so that…

Q: 1. Verify that the Mean Value Theorem applies to the function f(x) = v25 – x² on the interval [-3,…

A: Use mean value theorem

Q: 8. Find all the points in the interval [1,5] for f(x) = -x² + 6x – 6 such that the function verifies…

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Q: Let f denote the function if x < 2; 1 f(x) = if x 2. Which of the following is true about f? Of is…

A: Given  f(x)=x24if x≤22xif x&gt;2

Q: 6. Use definition of continuity to determine the value(s) of k such that the 2х + 5, х< 3 function f…

A: Topic : Continuity

Q: Let the function h: R → R be bounded. Define the function f: R→ R by f(x) = 1+ 4x + x²h(x) for all…

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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&lt;-24-x2     ;  if -2≤x&lt;2x-3  ;  if x&gt;2 We have to…

Q: snip

A: A function f is decreasing on an open interval I, if for any choice of x1 and x2 in I, with…

Q: Let f(x) = (|x| + x2 − 1     if x ≤ 0          (|x| − x2 − 1     if x > 0 Is there some x in (−1, 1)…

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Q: (a) Assume that f(x) is a function defined by f(x) = x²-3x+1 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x)…

A: Given.

Q: Let fn f, where if 0<x<! if <a<i -x if 0 <x<1 fn (x) n-2 (2 - x) 2 and f(x) %3D if x = 0 Then lim…

A: 2

Q: Let f(x)=x(4√x −3) Determine the absolute maximum and minimum values of f on the interval [0, 4]

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Q: Suppose f and g are real-valued functions defined on all of R, f is measurable, and g continuous. Is…

A: Given, that f and g are real valued function on ℝ and f is measurable and g is continuous function.…

Q: Prove that if (c, f(c)) is a point of inflection of the graph of f and f" exists in an open interval…

A: It is given that (c,f(c)) is a point of inflection of the function f.

Q: Let f: R R and let f(x) >0 for all xER. Show that f(x) is strictly increasing if and only if the…

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Q: Prove that if f (0) = g(0) and f' (x) ≤ g' (x) for x ≥ 0, then f (x) ≤ g(x) for all x ≥ 0. Hint:…

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Q: 9. If f (x) = prove that f (xr) is an even function. et -1 2

A: A function f(x) is said to be even if we have f(-x)=f(x)

Q: Find the constant a such that the function is continuous on the entire real line. 8sin x if x < 0…

A: For continuity of f(x) LHL=RHL     so find lhl and rhl and equate both to get a.

Q: Let f(x) = x³ – 2x2 – 4x + 2 on I = [-2, 2]. Verify that f satisfies the conditions of Rolle's…

A: Rolle's Theorem1.f(x) is continuous in the interval [-2,2] because it is a polynomial function2.f(x)…

Q: Suppose a function f is continuous on the interval [1, 5] except at x 3 and f(1) = 2 and A5) = 7.…

A: In this question we have to given two graphs one is holding IVT while another one is not

Q: 8. Let f,g be differentiable on R and suppose that f(0) = g(0) and f'(x) 0. Show that f(x) 0.

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Q: Find the value of that will make the function f(x) continuous everywhere. f(x) = {ka² -2, (4x+k,…

A: Given function is- f(x)=4x+kx≤-5kx2-2x&gt;-5 To find- To find the value of k that will make the…

Q: Prove that f(a) = (z - 1)² + (x + 1) is an even function.

A: The given function,   f(x)=x-12+x+12

Q: Does there exist a function f such that f(0)  =-1, f(2)  =4, and f'(x) ≤ 2  for all  x?

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Q: Let f(x) be twice differentiable and invertible function such that f'(x) >0 and f"(x) > 0, prove х,…

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Q: Show that g '(x) is piecewise continuous for all x, on the interval -L x

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Q: Assume that f(x) is a function defined by x²-3x+1 f(x) = 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x) is…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Prove that f(x)=x^3+x-1 has a root in the interval [0,2]

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Q: Define g(-1) in a way that extends g(x) = 1 to be continuous at x= - 1. x² -x-2 2 9(-1)-D

A: Given that: g(x)=x2-1x2-x-2

Q: let f: R^ -> R be a function given by 2 f(x₁, x₂... x₁) x.x²2. x²n, where = € 2.1 Show that the…

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Q: Find the values of a and b that make f(x) continuous for all real numbers. (x+2, f(x) = { ax³, (x² +…

A: Given that f is a continuous function, then it is continuous at all the point. So, the function is…

Q: 1. Let f(x) = 4x − 4x 2 . Verify that f(x) satisfies the conditions of Rolle’s Theorem on the…

A: Note:- According to the company policy we can attempt ?️ only one question so please upload the next…

Q: Let f(x)=1/x. Show that f is bounded from above on any interval (α,1), where α>0, but the function…

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Q: Letf(x)= 1/x-1 Use a 8-e proof to show that f is continuous at x = 2.

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Q: III. 1) Show that F(x) = x+4 is continuous on (1,5) 2) Determine the largest interval over which…

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Q: Verify that the function f (x) = x² – 8x + 15 defined over the interval [3, 5], satisfies Rolle's…

A: using role's theorem: if a function is continuous and differentiable on the closed interval [a b]…

Q: Find all values of a such that the function f(x) = (x²+ 2x, if x<a -1, if x2a is continuous…

A: Given: x2+2x,    if x&lt;a-1,      if x≥a

Q: Find the values of a and b that make f(x) continuous for all real numbers. (x+ 2, f(x) = { ax³, (x²…

A: f(x)=x+2,x≤-1ax3,-1&lt;x≤1x2+bx,x&gt;1

Q: Let f(x)=x^3-2x^2-4x+104 on the interval [0,3]. A. Find all crirical numbers of F(x) inside [0,3]…

A: We have to calculate absolute maximum and minimum.

Q: Show that a function f(x) = ∑ ∞n=2 (1/x-nπ + 1/nπ) is well defined on [0, π]. That is, prove the…

A: Given function,

Q: Suppose f(x) is a smooth function that passes through the points (3, –5) and (-6, 4). Prove that…

A: Given,        Suppose fx is a smooth function that passes through the points 3,-5 and -6,4.…

Q: (5) Let f: RR be twice-differentiable f(x) ≤0 and Prove that f is constant on R. on R. Suppose that…

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Q: The function of f is increasing on I if ______ when x1 < x2.

A: We have to find when function is increasing

Q: rove that there does not exist a function f € L'([0, 1]) such that for any integer n > 1, f(x) >…

A: Let f∈L10,1. That is there exists sequence in 0,1 such that fxn converges. Since the sequence fxn…

Q: Show that f is a discontinuous function for all x, where f (x) is defined as follows: Show that f 2…

A: Here given function as fx=1for x rational-1for x irrational Also  f2x=fx·fx

Q: Let f(x) be the function defined by f(x) = x3 -1 for x < −1, f(x) = ax + b for −1 ≤ x ≤ 4, and f(x)…

A: Given f(x) x3-1   for x&lt;-1        ax +b  for -1≤x &lt;4        x2+2 for x≥4