nx 4. Let fn(x) = 1+7²², x € R. Prove that fn → 0 pointwise on R, then check whether fn → 0 uniformly or not on [0, 1]

Q: x(x²+3) Q7. Consider the function f : [0, 1] → R, → √√4+24 Explain why (a) f continuous, (b) f is…

A:

Q: Let f be a function from N to (0,1). Prove -h for any nEN, y = f(n). Using part (a), prove that…

A:

Q: Prove that x² + ¹² +2111²2 = y 1112 where prime denotes differentiation w.r.t.s. 1 1+p¹² 4 p² o² + P

A:

Q: 2. Let f:Z→Z be the function given by fAn) =3n+1. a. Is fone-to-one? Briefly explain why. b. Is…

A:

Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

Q: Let f(r) = In r Find and simplify f"(x).

A: Please refer the attached image for complete solution. THANK YOU.

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?

A:

Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

A:

Q: 1 If the function f is continuous on [a,b], then there exists at least one number C in [a,b] such…

A: If the function f is continuous on [a, b], then there exists at least one number C in [a, b] such…

Q: 3c. Let 0 < a < 1 and ƒ : (a, 2) → R be given by ƒ (x) = ½. Show that ƒ is uniformly continuous.

A: Given: Let  and  be given by We have to show that  is uniformly continuous.

Q: Prove that F (3.x)(A → B) → (Vx)A → (3x)B. 3.

A: To prove: ⊢∃xA→B→∀xA→∃xB

Q: 3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f…

A:

Q: Let f: Nx N → Z defined by f(n, m) = n − m² 1. Find f({(0, 0), (1,0), (0, 1), (1, 1)}) 2. find…

A:

Q: Let h : R → R be some twice-differentiable function. Define f(x,y) = h(x) for all x ≠ 0. Show that…

A:

Q: Prove f(x) = x2+4 is not uniformly continuous on R.

A:

Q: 3. The function f(m, n) = 2"3" is a one-to-one function from N x N into N. (a) Calculate f(m, n) for…

A:

Q: Show that the function f: N² → Z given by f(m, n) = m²-n² is neither injective: suriective. nor

A:

Q: If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly…

A: