2. Show that the following functions are convex by verifying the condition that V²f(x) 20 is satisfied for all a in the domain of f: (a) f(u₁, u₂) In(e": +e"), (b) f(u₁, U2, U3, U4) -In(1-₁-₂-3-4) over the domain {u E R¹|u₁ + ₂ + 3 + 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: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

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Q: Let f: R³ R be the function defined as f(x, y, z)= √√₂² + y² + 2², 0. if (x, y, z) = Q³, otherwise.…

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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 =…

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

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Q: 2. For F(x, y, z) = (6xsin y+10x=' )i + (3x² cos y-4ye* −2y²z³)j + (-2y³e² +15x²2²)k, find a…

A: Given Data: Let us consider the given data, Fx,y,z=6x sin y +10xz3i+3x2 cos y -4yeyz-2y2zeyzj…

Q: (c) For any (a, y) E R?, let F(x, y) = (4x - 3y + 1, 3x +4y + 1). Then F is a similarity.

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

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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: 5. Express the given function using unit step function: * [t f (t) = 0st<4 2 , t24 O A.) f (t) = t…

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Q: Determine the largest subset S of C where the function f(z) = 3x2 + 2x - 3y2 - 1 + i(6xy + 2y) is…

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Q: Let f : D → R be defined by f(x) = (x+1)2/x for x ∈ D (D ⊆ R is the largest possible domain of the…

A: The domain of the function  f:   Domain: The domain of a function is the set of input or argument…

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

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Q: Let f : Z+ → N be the floor function f(x) = [(x – 1)/2].

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Q: Consider the 2-dimensional function f(x, y) = xy²exp (-³² + y). a) Compute all the stationary points…

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Q: 1. Let D = the sales of DVDs in billions of dollars, S = the total sales of streaming videos in…

A: The sales of dvds given as the function D = -0.92t + 8.75 and the total sales of streaming videos…

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: Suppose that f(x, y) is a differentiable function of two variables defined on the entire xy–plane,…

A: Let us consider a function f:ℝ2→ℝ defined by  fx,y=x2y2                                      (i)…

Q: Determine whether cach of these functions from Z to Z is one-to-one. a) f(n)=n- I c) f(n)=n b) f(n)…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…

Q: 6. Let S = {-1,0, 2, 4, 7}. Find f(S) (image of S under f) where f is from R to R for (a) f(a) = 1…

A: The

Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

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Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: 1. Find the following for the function f(x, y) = 3x³y² - 5x + 26y - 9. a. f(1,2)= b. fy(x, y) = c.…

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Q: 2. Let f: [0, o0) R be the function given by f(r) x2 + 2. Is the statement Vrvy((f(x) > f(y)) → (x >…

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Q: 18. Let A = {1,2, 3}, B = f: A → B, and g : B → C, such that {W, Y, Z}, C = {7, V2, V5}. Find…

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

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Q: 6. For the following functions, show whether they are concave/convex/neither and then show whether…

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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 f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

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