Question: Let f : R - R be a continuously differentiable function and suppose f'(x) # 0. Show that there exist neighbourhoods U and V of

Let f : R - R be a continuously differentiable function

Let f : R - R be a continuously differentiable function and suppose f'(x) # 0. Show that there exist neighbourhoods U and V of x and f(x) respectively, such that flu : U - V is bijective. Moreover, fis differentiable at f(x) and its derivative equals 1/f'(x)

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Exercise 4.4.7: a) Let f : R - R be a continuously differentiable function and k > 0 be a number such that f' (x) > k for all x E R. Show f is one-to-one and onto, and has a continuously...

3. Let f : R R be a continuously differentiable function and let f'(-) denote the derivative of f. Using only the definitions/theorems we discussed in class, prove that: a. If f'(x) > 0 then f is...

1. Let X R be a compact rectangle and let [a, b] R be a compact interval. Let U R+1 be an open set such that XX [a, b] U and let f: U R be a continuously differentiable function on U. Define the...

Question 4 (15 marks, C3) (a) Let f : R - R be a continuously differentiable periodic function of period 27. Given that the Fourier series of f(x) is 2 413 - k - sin kx. K=1 Moreover, 1 572 k=1 (41-3...

Let f : [0,) R be a continuously differentiable function such that f(0) > 0 and there is a constant C with |f 0 (x)| C 0. (b) Prove that...

2. (5 marks) Let f : R - R be a twice differentiable function such that f"+ f =0. Prove there exist constants c and c2 such that, for all real x, f(x) = ci sinx + c2 cosx. We are assuming here that...

7. Let f (x) be a twice continuously differentiable function on an interval a, b], and suppose n is a natural number. The notes define (without any explanation) the Simpson's method approximation of...

Q1 (10 points) Use theorem 11.9 to show that the function f (x1, x2) = x x2 , d 1/2 1/2, defined on R2, is concave. Theorem 11.9 Let H be the Hessian matrix associated with a twice continuously...

1. Consider the initial value problem for the autonomous system x' = f(x), x(0) = am, where f : R\" > R" is a continuously differentiable ftmction. (a) [Bonus] Show that if ](x0) = (m) with 3

Let f : R n R be a differentiable map and let F : R n+1 R be given by (x, z) 7 f(x) z, where x R n and z R. Let M R n+1 be the graph of f and let N = F 1 ({0}). (a) Show M = N. (b) Let a R n ....

What are financial frictions, and what role do such frictions play in the financial crisis? How does a financial friction enter the IS/MP diagram and the AS/AD framework?

What is the de Broglie wavelength of

Investor Pitch A business plan is normally the first - if the reader doesn t like what he / she sees, the only contact between entrepreneurs trying to build a business and potential suppliers of...

An aqueous solution contains 2.8 mM of total ions. Part A If the solution is NaCl(aq), what is the concentration of chloride ion? Express your answer as a concentration to two significant figures. 15...

The town of Hillside has a surface reservoir that supplies water to the homes in the towns central residential district. A committee of residents is concerned about the systems capacity and wants to...

You work as a freelance accounting professional and have been recently engaged by the auditors of Life-Positive Inc. for an assignment. In examining the company's records, you have extracted the...

After reading the companion article ( Viewing Brands in Multiple Dimensions ) and reviewing the companion video (What Consumers Want), provide an executive briefing note regarding the significance to...

The Pacific Digital Cellular (PDC) TDMA system uses a 42.0 kbps data rate to support 3 uses per frame. Each user occupies 2 of the 6 time slots per frame. (a) What is the raw data rate provided for...

Create a 200 word summary on the Juvenile Justice System than and today. Describe The Juvenile system and provide a generalizable definition. Highlight the issues in the Juvenile system and why this...

(Two Differences, 2 Years, Compute Taxable Income and Pretax Financial Income) The fol- 3,5, lowing information was disclosed during the audit of Munter Inc. 2. On January 1, 2006, equipment costing...

(Five Differences, Compute Taxable Income and Deferred Taxes, Draft Income Statement) King = _ Company began operations at the beginning of 2007. The following information pertains to this company. ;...

(Explain Future Taxable and Deductible Amounts, How Carryback and Carryforward Affects Deferred Taxes) Mary Joe Fernandez and Meredith McGrath are discussing accounting for income taxes. They are...