Q. 7. Let f (x) be derivable over [a, b] such that f' (a) = 10 and f'(b) = 20. Prove that the re exist c E (a, b) such that f' (c) = 15.

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A: Given: Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). To Find:…

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A: our objective is to find the required derivative

Q: Let f:A-B and X,Y be subsets of B. Then f-¹(XnY) = f'(X) nf-¹(Y). True False

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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. Let f:Z→Z be the function given by fAn) =3n+1. a. Is fone-to-one? Briefly explain why. b. Is…

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Q: suppose that f (u, v) = ue", where u =rs and v =r + s. The composite function f(u, v) is equal to:…

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Q: Consider the function f(x)=1/x+4. a) As x→−∞, f(x)→ b)As x→∞, f(x)→

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Q: Consider the functions f(t) = e' and g(t) -81 defined on 0 st<o. %3D = e (a) (f* g)(t) can be…

A: Consider the functions, f(t) = et and g(t) = e-8t

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Q: . Let f : [0, 1] R be defined by |-1, z = ; >50 f(x) = {0 |<a < }; 1, 21 (a) Show that f € R([0,…

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Q: Suppose that f'(x) > M for all x E [a, b]. Prove that f(b) > f(a) + M(b

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Q: Let f(2)= iz³ + cos(z²) z² +1 i. Find f'(z). ii. Evaluate ƒ'(–2i) and simplify. iii. State the…

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Q: Suppose f, g:C → C are differentiable. Prove that fg is differentiable and that (fg)' = f'g + fg'.

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Q: When z = f(x,y), which of the following is true for the whole statement? O ddz = f"(x).dx² and dx2…

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Q: Let /: X →Y be a function, A, BC X and C, DCY. Show that: 1. f(AUB) = f(A) uf(B) 2. If AC B, then…

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Q: Find f(2021) if function f satisfies following conditions: f(1)=1, f(x+y)=f(x)+f(y), for all x,y…

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Q: 4. Find f"(2). (a) f(u) = 2u²5 – u).

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Q: 7. Prove F (Vx)(A → B) → (3x)A → (3x)B. 8. Prove E (¥x)(A → (B = C)) → ((¥x)(A → B) = (¥x)(A → C)).

A: Given: Let us prove that  7.↦∀xA→B→∋xA→∋xB and 8.↦∀xA→B≡C→∀xA→B≡∀xA→C

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Q: for If f (2) is a regular function of z, prove that f (2) 2 = 4 | f'(2) 1?. %3D ve I f12) 1? =4 |…

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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: 15. Let f(x, y) = 22 if (x, y) ‡ (0,0), and f(0,0) = 0. Is ƒ continuous at (0,0)? x+y

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