Suppose, for j = 1, 2,... ,n, that fj are real functions continuously differentiable on the interval

Question:

Suppose, for j = 1, 2,... ,n, that fj are real functions continuously differentiable on the interval (-1, 1). Prove that
g(x) := f1(x1) ∙ ∙ ∙ fn(xn)
is differentiable on the cube (-1, 1) × (-1, 1) × ∙ ∙ ∙ × (-1, 1).

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

An Introduction to Analysis

ISBN: 978-0132296380

4th edition

Authors: William R. Wade

See More Books

Question Posted: Mar 29, 2016 04:19 AM

See More Questions

Suppose that (a, b) is an open interval, that f: (a, b) R is differentiable on (a, b), and that x0 (a, b) is a proper local maximum of f (see Exercise 4.1.8). a) Prove that given > 0, there exist...
Suppose that I is a closed interval and x0 I. Suppose further that f is differentiable on R, that f'(a) 0 for some a R, that the function satisfies F(I) , and that there is a number 0 a) Prove that...
Suppose that f: Rn R is differentiable at a and that f(a) 0. a) Show that for ||h|| sufficiently small, f(a + h) 0. b) Prove that Df(a)(h)/||h|| is bounded for all h Rn{0}. c) If T := -Df(a)/f2(a),...
? ?? ? Prepare Company 1's COMPLETE CASH FLOW STATEMENT for 2016 (that includes CFO, CFI, CFF) using INDIRECT approach for CFO part. Explain how to treat for accounting purposes occupation of...
Fondren Exploration Ltd. has 1,000 convertible bonds ($1,000 par value) outstanding, each of which may be converted to 50 shares of stock. The $1 million worth of bonds has 25 years to maturity. The...
In 1971, after the Swann v. Charlotte-Mecklenburg Board of Education decision that permitted busing to achieve racial integration, schools were required to become integrated immediately.* In...
Consider a randomized pair design with $n$ units where two treatments are randomly assigned to each unit, resulting in a pair of observations $\left(X_{i}, Y_{i} ight)$, for $i=1, \ldots, n$ on each...
The combined gas mileage of midsize cars varies with mean 24 miles per gallon (mpg) and a standard deviation of about 5.5 mpg. A particular rental car agency typically has 150 midsize cars in its...
A thin coil at the origin of the coordinate system has 1 0 0 turns, radius 1 . 4 cm and current 2 A . the magnetic field due to the coil at points on a circle in the xy plane with radius...
a) Comment on factors likely to affect the accuracy of the analysis of costs into fixed and variable components. (b) Explain how the analysis of costs into fixed and variable components is of use in...
Prove that a) b) c) d) e) L{1}(s) =-, s>0 Lie" }(s) =-a, , S a, a E R Cicos(b))>0, beR L{sin(bt))(s) s > 0, b E R s2+b2.
Let V be open in Rn, a V, and f : V Rm. a) Prove that Duf(a) exists for u = ek if and only if fxk (a) exists, in which case b) Show that if f has directional derivatives at a in all directions u,...
Define net working capital. Explain how increasing the amount of trade credit can reduce the net working capital for a company.
Which of the following is a type of plasmid? a. F factor (fertility factor) b. R factor (resistance plasmid) c. Virulence plasmids d. All of the above are types of plasmids.
Which of the following is not a feature of the DNA double helix? a. It obeys the AT/GC rule. b. The DNA strands are antiparallel. c. The structure is stabilized by base stacking. d. All of the above...
In the lagging strand, DNA is made in the direction _________ the replication fork and is made as __________. a. toward, one continuous strand b. away from, one continuous strand c. toward, Okazaki...
During transduction involving a P1 phage, a. any small fragment of the bacterial chromosome may be transferred to another bacterium by a new phage. b. only a specific fragment of DNA may be...
The function of transposase is a. to recognize inverted repeats. b. to remove a TE from its original site. c. to insert a TE into a new site. d. all of the above.
The probability of an event can be larger than 1 provided that you are absolutely sure that the event will occur. Determine whether each statement is true or false, and explain why.
The following items were displayed in the statement of affairs for Lubbock Company: Fully secured liabilities ......... $90,000 Partially secured liabilities ....... 12,000 Unsecured liabilities...
Consider the simple linear regression model Y = B0 + B1x + , with E() = 0, V() = 2, and the errors uncorrelated. (a) Show that E(2) = E(MSE) = 2. (b) Show that E(MSR) = 2 B1SXX.
Suppose that we have assumed the straight-line regression model Y = BO + B1X1 + but the response is affected by a second variable x2 such that the true regression function is E(Y) = BO + B1X1 + B2X2...
Suppose that we are fitting a line and we wish to make the variance of the regression coefficient B1 as small as possible. Where should the observations xi, i = 1, 2, p, n, be taken so as to minimize...
Jorgansen Lighting, Incorporated, manufactures heavy-duty street lighting systems for municipalities. The company uses variable costing for internal management reports and absorption costing for...
Assume that the stock of South Gate Corporations sells for R108 and is expected to increase by a factor of 1.25 or decline by a factor of 0.83 in 10 months. As a portfolio analyst, you find a call...
Use the below information to calculate the profit or loss of a short call butterfly spread if the spot price of the underlying asset at maturity is R89.45. June 70 Call costs R1.18 June 70 Put costs...

Suppose, for j = 1, 2,... ,n, that fj are real functions continuously differentiable on the interval

Question:

Step by Step Answer:

Let V denote the open cube 1 1 1 1 Clea...View the full answer

An Introduction to Analysis

Students also viewed these Numerical Analysis questions