-2 x<-1 Q1)a) For what values of a and b is g(x) = |ax - b -1

Q: 1. Show that the function, defined for x E R by f(x) = 1 is uniformly continuous 1+x² on R.

A:

Q: Show there does not exist a continuously differentiable function f on [O, 2] such that f(0) = -1,…

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

Q: Show

A:

Q: 2. Show that f(r) = sin r is uniformly continuous on (0, 00). %3D

A: 2. We need to prove that sinx is uniformly continuous on 0,∞. Definition of uniformly continuous…

Q: 17. Show that f(x) = x² +2x is uniformly continuous on the interval [0, 10], using the e -8…

A:

Q: Does there exist a continuous function f(x) such that f(0) all x in (0, 2)? 10, f(2) = 2 and f'(x)…

A: Let's find.

Q: 1. Let the function f: R R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Differentiablity analysis problem....

Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,... then show that f(x)=0…

A:

Q: ɔ that h is a continuous functi (x2 — 2, х < 6 h(x) 3= 7х + a, х 6

A:

Q: 5. If f is continuously differentiable on (0, 1] and f(1) = f (0) = -1/6, prove that dr 22 2 Hint:…

A: Continuously Differentiable Function: A function f is continuously differentiable on [0, 1] if its…

Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

Q: Suppose f is differentiable on some open interval containing e and f" (c) = 0 then (c, f(c)) is a…

A:

Q: Q1/ Discuss the continuity of : 1 x+ for x 2

A: Since, you have post multiple question. I am answering first one. Please post again for remaining…

Q: 4 Given that f(2) is analytic at z = 0 and that f G) =, n = 1,2, 3..,find f(2). VE

A:

Q: 2. If f is continuous on an open interval containing x, and fchanges its direction of concavity at…

A:

Q: . Apply Jensen's inequality (2) to f(x) = In x, with i 1 = ..= d = 1/n to prove that 1 Vx1x2...X 0,…

A: Given :-  Apply Jensen's inequality (2) to f  x  = ln x  , with λ1 = λ2 = ..........  = λn = 1n to…

Q: lim, - f{x)= 0, then f(x)dx diverges Select one: O True False

A: When the limit exist or equal to 0 then the series is convergent otherwise divergent.

Q: (b). Show that iff= u+iv is analytic in a region S and u is a constant function (i.e., independent…

A:

Q: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: a) f(x)dx = 0 -- b) f?(x)dx = 2 | f²(x)dx -h

A:

Q: 3. Consider the function g(x) defined by | 22 sin(1/x) if x > 0, g(x) = –x² if x < 0. (a) Show that…

A:

Q: The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that…

A: g(x) = 0 for 7 distinct values of x in (a, b).Drawn a rough sketch of the case

Q: Consider the function f(x) = 5 – 5x2/3 on the interval [-1, 1]. Which of the three hypotheses of…

A:

Q: 7. Prove that if the functions f and o are bounded and integrable in [a, b], then 6. 9. f(x) $(x) dx…

A:

Q: how (in terms of  − δ) that a function f : R 3 → R defined by f(x, y, z) = (2x + 3y + 4z) is…

A: Solution of given problem is followed by

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

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

Q: 1. Show (in terms of  ε - δ) that a function f : R3 → R defined by f(x; y; z) = (2x + 3y + 4z) is…

A:

Q: 18. Find numbers a and b so that r+ 4, ux + b, 3r 8, x> 3 1<r 3 is continuous everywhere.

A: we need to find a=?                          b=?

Q: x + 3, x 2 x - 2

A:

Q: Σ sin k+1

A: to show that the given series ∑k=1∞1ksinxk+1 converges uniformly on R to a differentiable function f…

Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

A:

Q: The function f(x) interval [1, e] at x = Hora lo 0 x is the nat

A:

Q: 2. Suppose that f is a measurable function satisfying m({r €R: [f(x)| 2n}) < ne N. n3' Is f…

A: Given that f is a measurable function, with the following condition satisfied. m(x∈ℝ:…

Q: Q\ Use appropriate functions to guarantee that (). x sin if x # 0 h(x) = 0, if x = 0 is continuous…

A:

Q: Show that the converses of parts (a) and (b) of Theorem 6.2.8 are false by finding counterexamples.…

A: Theorem: Let f be a differentiable function on the interval I. Then, If f'x&gt;0 for all x∈I, then…

Q: A function ƒ is continuous for all x and has a local minimum at (2, –5). W statement below must be…

A:

Q: Q-2-for what valne of b is x+2b gex)=- 2 x²_b b-3 2. Continuous at every x ?

A:

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

A:

Q: 3 - x, x<2 Q.No.15: Is f(x) = {+2 x22 continuous at every x € R. x+1 xs1 -.

A: Continuity of a function.

Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

A:

Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

A:

Q: 2.23 Let 1 - x2 f(x): if x € Q, if r ¢ Q. Find all points p at which f is continuous. Prove your…

A:

Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

A:

Q: yLet f(2) be analytic in {z: |2| < 1} with f(0) that |f"(0)| < 2. = 0 and |f(2)| < 1 for all z in…

A:

Q: Assume thatf is differentiable in (-oo, x) and af'(r)N 0 for all a E (-00, 00). Prove that f has the…

A:

Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.

Q: 2. Suppose that f is a measurable function satisfying m{r ER: |f (x)| > n} ) < 1 nEN. n3 Is f…

A:

Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: Determine where h(x) = sin(x- Sx + 2) is continuous. %3D h is continuous on the interval [0, co). h…

A: