If |fn(x) – fn(y)l < M|x – y|ª for some fixed M and a > 0 and all x, y in a compact interval, show that {fn} is uniformly equicontinuous.

Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: Let f(z) be an entire function satisfying that f(z)| ≤|z² for all z. Show that f(z) = az² for some…

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Q: Evaluate |[[p(4z + 3y² + 2x)dV = {(x, y, z) | 0<x < 2,0 sys 2, 0 s z < 1} .

A: We have to evaluate the triple integral given below: ∭B4z3+3y2+2xdV  Here, the region B in three…

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

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

Q: Let f [0, 2] → R be a continuous function, with ƒ(0) ≤ f(1) ≤ f(2). Show that there : exists xo [0,…

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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: 4. Prove that f(x, y, z) = 2 - 1 x² + y² is continuous in all its domain.

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Q: Let's assume that f(x)= r²+c* I c-16 g(x, a) = a* x² + ar Find a such that the following error is…

A: Given that where            That is  Also, given that

Q: [1+2p(x-1)+(x-1)² x≤1 x>1 [qx+p -) -) Find the values of q, in terms of p, for which fis continuous…

A: Given:f be a function defined by:a. To find the value of q in terms of&amp;nbsp;p, for which f is…

Q: Find the absolute maximum and minimum values of f on the set D. f(x,y) = x2 + y2 + x2y +6 D =…

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Q: Find the absolute maximum and minimum values ​​of f (x, y) = x² + 2y²-x in the set D = {(x,y) ∈ R²…

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Q: Find the absolute maximum and minimum values of f on the set D. f(x, y) = x4 + y4 − 4xy + 6, D =…

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Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

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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: 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…

Q: Show that if f"(x) < 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] < favg < f[(a+b)/2]

A: Consider the given information. f"(x) &lt; 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] &lt; favg &lt;…

Q: -2 x<-1 Q1)a) For what values of a and b is g(x) = |ax - b - 1<x< 1, 3 x21 Continuous at x=1

A: From  the given problem:   gx=-2x≤-1ax-b-1≤x&lt;13x≥1     As we know that a function f(x) is…