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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

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

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.2: Integration By Parts
Problem 44E
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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 Vx ER Prove that f(x) ≥ ƒ(0) + cx Prove that f is surjective. That is, ƒ(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
Transcribed Image Text: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 Vx ER Prove that f(x) ≥ ƒ(0) + cx Prove that f is surjective. That is, ƒ(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
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