Show that the following functions are convex by verifying the definition, i.e., that. f(x + (1-A)y) ≤ f(x) + (1 - A)f(y) is satisfied for all x, y in the domain of f and all A = [0, 1]: (a) f(u) = 1,u> 0, (b) f(u) u, u € R. =

Q: 6. Determine the type of the following functions ( Injection, Surjection, Bijection?). a) Let f be…

A: The solution given as,   a:-  Let 'f' be the function from             a,b,c,d to1,2,3      and…

Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

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Q: 3.23 Prove or give a counterexample: The function |f| € R[a, b] + ƒ € R{a, b].

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Q: Check whether Rolle's theorem conditions are satisfied or not for the given function f(z) =z- 4z on…

A: The given function is: f(x)=x2-4x Now, the conditions of the Rolle's theorem is: 1) f is continuous…

Q: 2. Show that the following functions are convex by verifying the condition that V² f(x) 20 is…

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Q: 1. Let h : R → R be some twice-differentiable function. Define f(x,y) =h() for all x ≠ 0. Show that…

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Q: Let f: S¹→ S¹ be a continuous function. Proof or disprove the following statements : a) f has a…

A: Given that the  function f : S1 → S1 be a continuous function  where S1 = x,y∈ℝ2 : x2 + y2  =1 We…

Q: Exercise 4. Let F and G be homothetic functions in two variables. Is their sum homothetic? [Hint,…

A: We will use the fact that the MRS of a homothetic function is always homogeneous of degree 0. Let…

Q: Which one of the following relations is not a function? O ((x.y): r, y ER and r + y² = 1) O ((x.y):…

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Q: Let f : R → R be a function defined by f(x) = Decide whether it is an injection, a surjection, 1+ x2…

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Q: Let f be the function defined by f (x) = x³ – x² – 6x. What is the absolute maximum value of f on…

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Q: Let h : R → R be some twice-differentiable function. Define f(x,y) = h(x) for all x ≠ 0. Show that…

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Q: Identify an inner function (call it g) of y = (5x + 4)3.Let u = g(x) and express the outer function…

A: y=(5x+4)3

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: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

A: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

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Q: 07. Suppose that f, g : R → R, are convex functions of one variable things. Let's define a function…

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Q: Let f: X --> Y be a function, and let U be a subset of X.  (1) Prove or disprove that f maps X \ U…

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Q: B. Let f: R²R be a continuous function, and let S be the subset of the R2 given by all pairs (x, y)…

A: Given: Let fx,y=x2+xy+3y2-3. To Show: fx,y contain minimum and maximum value.

Q: 2. Let us define a. Verify that f(x, y) = c. Now consider the function git Similarly, consider the…

A: Given Data: Let us consider the given function is fx,y=xyx2-y2x2+y2

Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

Q: Determine the domain and range of the function f: {r,s,t,u} ➔ {A,B,C,D,E} defined by f(r) = A, f(s)…

A: Domain of a function is the set of possible values of independent variable. The domain of given…

Q: Q1 a) let f be a function defined as f(x)=3x² + 2x². Find f(2), f(-1), f(1). b) Define functions f…

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Q: 2. Consider the following three functions: f : Z+ x Z+ → Z+ f(m, n) = mn h: Rx Z+ → Z+ h(r, y) = 7y…

A: