Let fn(x) on [0, 1]. 1+ x" (a) Prove that fn converges uniformly to 0 on [0, €], Ve E (0, 1). (b) Does fn converge uniformly on [0, 1]? Prove or disprove.

Q: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

A:

Q: 3. Let f be continuous on [-1, 4] and differentiable on (-1,4). If f(-1) = –1 and f(4) = 4, then the…

A:

Q: Let fn(x) = 1+cos2 2023 (na). Does {f} converge uniformly to a function f on R? √n

A:

Q: 4-x 2. s] Let f(x) = . It was computed thatf'(x) = (x² +4)z" Find where the function is increasing…

A:

Q: 6x2+5 If f(x) = In x4/5 find f' (x).

A:

Q: sume that f(t) is continuous on [a, b| and that u = g(t) is cont ferentiable on [a, b], then f(u)…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Let F(t) = (4t³ - 1)j + (5t+2)k. Find F'(t). 7"(t) O 288t 288t³j 05+288t³ O 36t³ +5 O None of these

A:

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

A: Since  f'(x)≥M therefore the function is differentiable and hence continuous let f is continuous on…

Q: Exercise 4: Suppose f is differentiable on (2,6) and continuous on [2,6. Suppose further that 1 < f'…

A: Given:f is differentiable on (2, 6) and continuous on [2, 6].

Q: 7 Find the most general function F such that F'(): - 5V?.

A:

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). • L₁ VE 4 + 4 dt Vt Step 1 F(x) = Note…

A:

Q: 6. Let f(x) = x2/3. (a) Then fis uniformly continuous on [-1,1]. Why? (b) Show that there does not…

A:

Q: Is g(x) = ln( x2+2 ) uniformly continuous on the interval (−∞,∞)?

A: given function is                                                        g(x)=lnx2+2 log is defined…

Q: ppose that f is a continuous function and the followi ƒ(2) = 2, ƒ(4) = −1, ƒ(6) = 6, ƒ(8) = = culate…

A: We have to choose the correct option from the given options according to the given information