X if x e Q If f (x) is defined on [0,1] by the rule f (x) = < (1-x , if x 4 Q Prove that fof (x) = x for all x e [0,1].

Q: 1. Show that the function, defined for x E R by f(x) = 1 is uniformly continuous 1+x² on R.

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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: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by f X,Y (x, y) = (x+y)/21…

A: Given that  The joint probability mass function of X and Y is  f(x,y)=(x+y)/21   , x=1,2,3 and y=1,2…

Q: 7. A function f : (a,6) → R satisfies a Lipschitz condition at x € (a,6) iff there is M > 0 and e >…

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Q: 8. If g(x) = e*In(x+1), compute g'(x). What techniques are you using in this case?

A: Solution is given below..

Q: 7. A function f (x) is continuous on the closed interval [2,5] withf (2) = 17 and f(5) = 17. Which…

A: Rolle's theorem

Q: If f(0) = 0 and f"(x) exists for all x>0, then show that %3D f(x) f'(x) xf"(5), 0 < § < x. Also…

A: Let's find.

Q: 2.if f (x) = x2 + 3, with domain R then f(x) has f(x) %3D O T F

A: Here the given function is f(x)= x2+3 with domain R We need to find it has inverse or not.

Q: a2f b- For the following functions prove that ax2 ay2 1- f(x,y) = e-2y cos 2x %3D f(x,y) = In/x2 +…

A: The first step inludes the  partial differentiation of the functions individually with respect to y…

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

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Q: 4- If a function is differentiable at zo, then it is : a) Differentiable on C b) Continuous on C c)…

A: To answer the given question about the complex function.

Q: of of and Find for the following function. f(x.y) = e xy In (2y) of dy

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Q: Let f(x)=x² -x² +x + 1 and g(x)= 3-х max{f(t):0sx; for 0 s x s 1} for 1 < x s 2 Discuss the…

A: Topic:- Functions

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: 1 Let f(x) = 8 + In (t) dt, for x2 1. Find f'(2).

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Q: 1) Consider the function f(x) = In(x + 5) + x + 5 Verify that the function F(x) = x In(x + 5) is an…

A: Explanation of the answer is as follows

Q: 4 Given that f(2) is analytic at z = 0 and that f G) =, n = 1,2, 3..,find f(2). VE

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Q: 2. Let f(x) = V4 – x for x < 4 and g(x) = x² for all x E R. (a) Give the domain of f · g, ƒ + g, ƒ º…

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Q: 8. Consider the function f(x)=/25-x² . a. Find the linearization L(x) of f(x) at x=3.

A: Given expression is , fx=25-x2 to find out the linearization L(x) of f(x) at x=3.

Q: +5, x <-1 Given f(x) = 3-x, -1< x s 1 x2+2, 1<x<2 X- Wx-2 + 4, x2 2 Find the point(s) where f(x) is…

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Q: 3 Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0 < |e'(x)| < |f'(x)| for all x €…

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Q: Let fnf, where if 0<a <! { fn(x) 교프2 (2-2) if <z<i if <x<1 n п-2 2 x if 0<x<1 and f(x) | %3D if x =…

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Q: If G(x,y) is differentiable with respect to x and y then it is always possible to find the number…

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Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

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Q: Зx2 — 1 if x 1 Determine c and d so that f is continuous everywhere

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Q: f (x) = x' - 4x + 2x – 3 %3D as x → -o0 f (x) → ∞ as x -0o f (x) → -0 as x →+of (x) as x +oo f(x) →…

A: topic - polynomials option - B

Q: Find the absolute maximum and absolute minimum of f(x) on the given interval f(x) = 2x2 – 4.x +3,…

A: To find the absolute minimum and maximum of a continous function on the closed interval [ a, b ], we…

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: 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: -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…

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Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

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Q: 29. Prove that f(x) =x² is uniformly continuous in 0 0, we can find 8 > 0 such that |x – x <e when…

A: to show f is uniform continuous on a set A , we need to show that for any given epsilon positive…

Q: b

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Q: Man of X = Avenge f= ELK) H.w prove that E (x-x) =0 Hint 又=E(x)

A: to prove EX-X=0 using the property of Expectation  EX+Y=EX+EY therefore, EX-X=EX-EX now let Xi be a…

Q: A function ƒ is continuous for all x and has a local minimum at (2, –5). W statement below must be…

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Q: For the function f(x)=sin(x)/x defined on [-1,1], there is some -1<c<1 such that f′(c)=0.f′(c)=0.…

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Q: (b) A function f AR is continuous at a point c E A if for any e > 0, unere exists A one 8 >0 but de…

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Q: II. Prove the following theorems on limits. 1. f(x) = L if and only if (x) – L] = 0 2. f(x) = L if…

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Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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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: a) Evaluate in terms of Gamma function ∫e^(−y^2) y^13 dy   limit 0 to ∞ b) Find ∫f(x) dx limit: 1…

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Q: For f(x) = √ √2-4 4- ™ Of is not continuous at x = 4 Of is continuous at a Of is continuous at a 5…

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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: If ø(x) =(x)+ f(1-x) and f"(x)<0 for all x e increasing in [0,1/2] and decreasing in (1/2,1].…

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Q: 1. Suppose that g(x) > 0 for all r, g(x) has a continuous derivative, g(-2) = 5 8 g(4) = 3 and |…

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Q: 2. The function (x) = x² is differentiable at x = 1. Find the function & such that o(x) = o(1) +…

A: Introduction: A technique for determining a function's derivative is differentiation. Mathematicians…

Q: 3 In x if x >1 2 2. Determine if f(x) = is differentiable at r = 1. x² +3 -1, if a < 1

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