Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing. Assume next that f"(x) exists for every x e (a, b), and prove that f is convex if and only if f"(x)20 for all x e (a, b).

Q: Suppose f : A → B is a function, X ⊂ A and Y ⊂ B. What does it mean to say: (a) z is in the range of…

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Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

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Q: Let f: [2, 7] → [0, ∞) be continuous in [2, 7] a (2,7). Then prove that there exists at least one…

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Q: Sketch the graph of a function f  that is continuouson [1, 5] and has the given properties.   f has…

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Q: 7 (:) Prove that f(x) : is convex.

A: Given the function f:ℝ→ℝ2 by, f(x)=xs Note that the domain of the function f(x) is convex.

Q: Prove that if f:X + Y is a one-to-one and onto function with inverse function f- : Y → X, then fof-l…

A: Given that, f:X→Y is a one-to-one and onto function with inverse function f-1:Y→X

Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

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Q: Let f e E(0, ox) and let a be a positive real number. Define the function fa as follows Sal) = {";…

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Q: Q2\A\ Which of the following functions are bijective 2- f: R{0}→ R\{0},f(x) = 1/x این 2- f:R→R,f(x)…

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Q: Find continuous functions f1, f2, f3, f4 defined on the open unit interval I = (0, 1) (i.e.…

A: 1. f1(x) = (M1 - L1)x + L1, for all x in (0, 1)  This function is linear and maps (0, 1) onto (L1,…

Q: 1. Let (a, bj be an interval in R and let be a continuous function on (a, b). (i) Check that the…

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Q: Let ƒ : R\{1} →→ R\{1} be defined by f(x) = x/(x - 1). Show that ƒ is bijective and find f-¹. Also,…

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Q: Let f be a function that is continuous on [1,7] and differentiable on (1,7). Suppose 2 < f'(x) < 3…

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Q: 7. has a fixed point. ) Prove that if f is a continuous function from [2, 5] to [2, 5], then f

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Q: Show that for any function f : X → Y , there are two functions g : S → T and h : X → S such that (a)…

A: To Show: Show that for any function f : X → Y, there are two functions g: S → T and h: X → S such…

Q: Show that f is continuous on (∞ , -∞ )  .

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Q: 13. Let the function f: [a, b] in R be continuous and one-to-one and such that f(a) &lt; f(b). Let c…

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Q: b) Let f: D→ C be a function and Y a subset of C. Show that f¹ [C-Y] =D-f¹ [Y].

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Q: How many functions f:R→R  exists such that : 2f(x)+2f(y)=f(x+y)+f(x−y)+2y holds for all real x and…

A: By choosing x=y=0. Also by choosing x=y we can see f(2x)=4f(x).  Moreover choose x=0 and use f(0)=0,…

Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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Q: 3. If f:(X,r)(Y.A) and g:(Y,A)→(Z,r) are open functions, prove that gof :(x.r)(Z.r) is open…

A: Definition: An open map is a function between two topological spaces that maps open sets to open…

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Q: 6. Let f(x) = x55x+1. Prove that f has at least 3 zeros in [-2, 2]; that is, there exist distinct…

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Q: b) Prove that if f(z) and f(Z) are analytic in a domain I, then I is a constant function.

A: Given Data: We have to prove  if fz and fz¯ are analytic in a domain Ω, then f is a constant…

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Q: Exercise 2. Let f : A → B be a function with Y1 and Y2 subsets of B. Prove that if Y1 C Y2, then…

A: Let f:A→B be a function and let Y1,Y2⊆B. Suppose that Y1⊆Y2 To show f-1(Y1)⊆f-1(Y2) Let x∈f-1(Y1)…

Q: If f is an analytic function and does not vanish in a domain D, then prove or disprove the…

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Q: Let f : R → R be an increasing function. Show that f(x) = x, ∀x ∈ R ⇔ f(x) = x, ∀x ∈ Q.

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