2. Show that the following functions are convex by verifying the condition that V² f(x) 20 is satisfied for all a in the domain of f: (a) f(un, u₂) In(e" + e"), (b) f(u, uz, us, UA) -In(1-₁-2-3-4) over the domain {u E R¹|u₁+uz+us+u₁ < 1}.

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: Are the following functions quasi-concave? Assume x ≥ 0 a) f(x) = a b) f(x) = a + cx², c < 0

A: Both questions are different so I soved 1st queation if you want 2nd one then repost it again.

Q: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

A:

Q: 1. Define the function f: C\ {-3} → C by 1 3+2 Compute (Ref) and (Imf) as functions of x and y,…

A: The given problem is to find the Re f and Im f for the given complex function. Given f : C \ {-3} →C…

Q: 2. Show that the following functions are convex by verifying the condition that V²f(x) 20 is…

A:

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. Consider a convex function ƒ :R +R. Recall that f is convex if f(tx+(1–t)y) 0. • Prove that for…

A:

Q: 21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁=1, x₂ = 16, x3 = 3, x4={, x3 =…

A:

Q: (#4) When is a function f(.): R" R convex and proper? Show that f(-) is convex and proper implies…

A: Function f : Rn→ R is convex if its domain is a convex set and for all x, y in its domain, and all λ…

Q: The domain of the function g(t) = vt – 2 is O R-{2} O [2, 00) O R о (-о, — 2) n (? m)

A:

Q: Let W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable, and the following applies.…

A:

Q: Let L(x) be the linearization of an arbitrary function f about c. Assume that f is concave down in…

A:

Q: Suppose that f : A → B is a function, C C B and D C B. Prove that Š (C – D) = } (C) - } (D).

A: Solution :-

Q: xMyn x2+y2 (х, у) 7 (0, 0) f(x, у) %D (х, у) %3D (0, 0)

A:

Q: Let f: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

A:

Q: If f(x)=x² -4, where x is an integer and 0<x<3, then which of the following sets of ordered pairs…

A: We have,

Q: 4. (a) Let f: X→Y, g: Y→ Z, and h: Y→ Z be functions. Prove or disprove: If gof=hof, then g = h. (b)…

A:

Q: 3. Determine whether each of these functions from is one-to-one/onto. (a) f : R → R, f(x) = 3r. (b)…

A: A function f is one-to-one if every element of the range of g corresponds to exactly one element of…

Q: The given functions are concave or convex or neither in the domain: 1). f(x, y) = 2r-y-r²+2ry-y² 2).…

A: As per the question we are given two multivariate quadratic equations : f(x,y) = 2x - y - x2 + 2xy…

Q: 2. Use the definition of convex functions to answer the following: (a) Show that f: RdR given by…

A: As per guidelines, I can do only first three subparts of the question.

Q: 3. If z = f(x, y) with the relation xy²z³ + x³y?z +1 = x + y+ z. Find dz at (1, 1, 1).

A:

Q: Find the absolute maximum and minimum values of f on the set D where f(x, y) = x² + 4y² − 2x − 8y +…

A:

Q: 4. Let X an Y be sets with A, B C X and let f: X→Y be a function. Recall that {f(x) EYE A}, and…

A:

Q: A function is defined as f(x,y) = 25xy + 3x³ - 7x² sin(y) aj ay a2f (ii) Find and 1325 3²x (iii)…

A:

Q: 2. For a linear function f: R" Rm, which of the following statements is/are true? Select all that…

A: Given , f: ℝn→ℝm be a linear function . (.)  A function  f:ℝn→ℝm is a linear function if , (i)…

Q: 3. Write the mapping rule for this function a) y = - 3f[-4(x - 1)] + 2: (x, y) → + d, ay+ b) f(x)=…

A:

Q: 5. Determine the vclue of c such that the function is continuous on (-o0,00 + flx) = 2x, x SC

A:

Q: Find the functionr that satisfies the following conditions. r'(t) = (1,4t ,5t4); r(1) = (7, – 6, –…

A: Given that

Q: F . dr, (c) Let f(x, y, z) = x³ + 2yª + 325 and F consists of three line segments from (1,0, 0) to…

A:

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

A:

Q: Prove that a function f: R→R is K-convex if and only if for any z ≥ 0, b > 0, and any y, we have…

A: To prove the statement that a function f:R→R is K-convex if and only if for any z≥0, b&gt;0, and any…

Q: 2. Let f= L(X,Y). Show that ƒ is one to one iff (f(x).(x)....(x)) is linear independent when (x,x)…

A:

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: 3. The function f(m, n) = 2"3" is a one-to-one function from N x N into N. (a) Calculate f(m, n) for…

A:

Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…