. If x, y E R with x < y, prove that x < tx + (1 – t)y < y for all t e (0,1).

Q: Let r(t): = 1 t-1' +-1/2). Find the domain of r(t). et+1, t-1,

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: Prove that f(x) = 5x2 + 3x – 26 is θ(x2)

A: Given: f(x) = 5x2 + 3x – 26 is θ(x2)

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

A:

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: Let A = R - {1} and define f: A→A by f(x) = (a) Find fo f(x). (b) Find fofo f(x). (c) Find f-¹(x).…

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Q: Suppose UCC is a domain, f, g = O(U). If |ƒ(z)|2 + |g(z)|² = 1 for all ZEU, show that ƒ and g are…

A: The solution to the problem is in the segment below.Explanation:To show that f(z) and g(z) are…

Q: 3. Let f: R³ R be defined by f(x, y, z) =|| x ||. Find Vf(x, y, z). What is the domain of Vf?

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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: Define briefly u(t) and 8(t)

A: u(t): u(t) represents the unit step function, which is defined as u(t) = 0 for t &lt; 0 and u(t) = 1…

Q: find the values of c that satisfy Rolle's Theorem

A:

Q: Let n > 2 be a natural number. Prove that there is no continuous function q : C → C such that…

A:

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: Suppose f(x, y) = xy-ax-by. (A) How many local minimum points does f have in R²? (The answer is an…

A:

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…

Q: g and go f, and the - 4, q(x) = x?

A: Is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)).…

Q: 7. ) Let f: N→ Z with f(n)=n+4. Let A = {1,2,3} and B = {1 € Z \ I <10). Use these to find the…

A:

Q: Verify that ||v-w||^2=1 when ||v||=2, ||w||=1, and v*w=2

A: By the definition of inner product and norm we can derive the following formula : ‖u-v‖2=⟨u-v,u-v⟩…

Q: For x, y E [0, 1], prove that 1 xy(1 – x – y) < 27

A: