. 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,

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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: Prove that x(t) = xoe At xoe At is unique solution

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Q: Prove that if Xn X and Ync, then (Xn, Yn) → (X, c).

A: Please see the answer innext step

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: (b) Show that the set A = {(x, y) E R² : y > 0} is not complete.

A: Given: The given set ,

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

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Q: Let x, y ∈ R such that x 0.001. Prove that |x · y| = |x| · |y|

A: Given: x, y∈ℝ, x&lt;-0.001 and y&gt;0.001   To prove: x·y=x·y

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: For x,y E [0,1], prove that xy(1-x-y)≤1/27

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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: B) If f(x.y) x'y? +x*y , find: axdy *Əyər

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

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Q: Let G = &lt;z | z6 = 1l and H = &lt;x, y | x2 = y3 =1, xy =yx&gt;. Showthat G and H are isomorphic.

A: Given :  G = z : z6 = 1 and  H = x, y : x2 = y3 = 1, xy = yx To show : G and H are isomorphic.

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: 2. Let g (x, y) = 2x² – y² . Compute 9(1, 2), g (2, 1), g (1, 1), g (–1, 1) and g (2, –1).

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

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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…

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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: Differentiate: f(x) = 4x³ - (5/x²) %3D O a. f(x) = 12x2 - (10/x³) O b. f(x) = 12x2+ (10/x³) O c.…

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Q: Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

A: f is increasing if first derivative is positive in the interval  and f has concave if second…

Q: If A%3 2x3 -Зх +2:х € [О, 2) then in (R, T,), A %3D а. [-3, 1] b. [-1, 1] с. [1, 3] С. d. [0, 4] e.…

A: Open sets and Closed sets: The sets in topology τ are called the open sets of the topological space…

Q: such that Let z = x+iy EC, for x, y E R. Find all analytic functions f Ref(2)= x² - y² − 3x + 4, and…

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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…

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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)…

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Q: a)Suppose f: R → Z, where f(x) = [x + ¹] i. Graph the function f(x). ii. If A = {x: 1 ≤ x ≤ 3}, find…

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

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Q: 1. For T € L(V,W), prove that null(T) = (range(T*)) and range(T) = (null(T*))+.

A: Given these, we can now prove the statements:  Proving (null(T)=(range(T∗))⊥):By definition, for any…

Q: Find all functions f:R → R that satisfy the equality xf(y) + yf (x) = (x+ y)f(x)f(y) %3D for all r,…

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