5. Suppose that f: R → R is a continuously differentiable function (of class C¹) and that there is a positive number c> 0 such that f'(x) > c VER Prove that f(x) ≥ f(0) + cx Prove that f is surjective. That is, f(R) = R. Prove that f is a diffeomorphism from R onto R. That is, f-¹ exists and is differentiable. (Hint: One can consider Mean Value Theorem, Intermediate Value Theorem and Inverse Function Theorem.) if x ≥ 0 and f(x) ≤ f(0) + cx if x ≤ 0

Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

A:

Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

Q: Let f be a continuous function defined on the closed interval [a,b] and differentiable on the open…

A:

Q: Let ƒ : ℝ → ℝ be continuously differentiable function and k > 0 be a number such that ƒ '(x) ≥ k for…

A: Let me show f is one one first,then onto then continuously differentiable....

Q: Let f:[-1,1]-R be defined by f(x) =x²sin- if x is not zero and f(0) =0, then .A f is differentiable…

A: When a function is one–one, i.e., when the values of the domain are equal, the value of the range…

Q: Let f(x) be a bounded, non-negative differentiable function on [0, ∞). Prove that the length of f(x)…

A: Given a function f(x) which is bounded,non-negative and differentiable on [0,)We have to prove that…

Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

A:

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

A:

Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

A:

Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)-f(x) f(y) ♥x, yER. Show…

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: Suppose f: IR→ R and g: R → R and h: R→ R are continuous functions such that h(x) = f(x)g(x) for…

A: Using given information we have to tell that whether the  given statement is true or false.

Q: Let f be a real-valued function on [a,b] that is differentiable on (a,b). Suppose f '(x) < 0 for all…

A: This is a problem of the application of differential calculus.

Q: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

Q: 2. T2: I can generate an example and explain why that example satisfies or does not satisfy a…

A: Since the their are multiple questions. So I am answering the first question only.for the other…

Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

A:

Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

A:

Q: Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one real…

A:

Q: Let f be differentiable on R. Suppose that f'(r) # 0 for every r. Prove that f has at most one real…

A:

Q: Prove the following. If f, g: [a, b] → R are continuous and both are differentiable on the open…

A:

Q: 4. Show that if f:[0,1]→[0,1] is continuous, then f has at least one fixed point. That is, there…

A:

Q: Suppose that f: R R is everywhere differentiable and that f'(x) 1 for all real numbers x. Prove that…

A:

Q: 4. Let f:R→ R be a function defined by f(x)=x for every xeX . Show that f is a continuous.

A: Let A ⊆ R , let f:A → R and let c ∈ A. We say that f is continuous at c if, given any number ε&gt;0,…

Q: If f(z) is an analytic function of z and f'(z) is continuous at each point within and on a closed…

A: We will use greens theorem and properties of an analytic function.

Q: Let f : I → J be a bijective differentiable function froman interval I to an interval J, and let F :…

A:

Q: Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

A:

Q: 4. Let A := (0, 1] and let f : A → R be defined by f(x) = !. Prove that f is continuous on A.

A: