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)

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).

Step by Step Solution

★★★★★

3.37 Rating (163 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

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

Question Has Been Solved by an Expert!

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

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

741-M-N-A-D-I (623).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis 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),...

# 5: Fourier Interpolant and DFT 1 The Discrete Fourier Transform (DFT) of a periodic array fj , for j = 0, 1, . . . , N 1 (corresponding to data at equally spaced points, starting at the left end...

Induction (#1-3, 24 pts) #1 . Prove by induction that8 + 15 + 22 + 29 + ... +(7 n + 1) = n (7 n + 9)/2for all n N . #2. Prove the summation formula for the following particular finite geometric...

Applied Mathematics and Computation 95 (1998) 181192 Love dynamics: The case of linear couples Sergio Rinaldi 1 Centro Teoria dei Sistemi, CNR, Politecnico di Milano, Via Ponzio 34/5, 20133 Milan,...

KINDLY PLEASE DIGITALIZED IT 11- Find the Area between the curve y=x(x-3) and the ordinates x=0 and x=5. Note: (hint) The Graph shows 2 regions. Get first the area of each region before adding it...

Kindly please digitalized the answer 1- Integrate the following. 1. fx'dx 2. JT 3. S(x2 - 2x + 5) dx 4. S(x + 2)5 dx 5. S1+ y2 2ydy 6. S cos(70 + 5) do 7. S ( x2 + 2x - 3)(x + 1) dx 11- Find the Area...

Note: All ML code must be explained clearly (INJAVAXX)and should be free of needless complexity. 2 CST.2016.1.3 2 Foundations of Computer Science Please help. (2c) (a) A prime number sieve is an...

George Bailey's capital at the beginning of the year was $14,000. At the end of the year his assets were $62,000 and his liabilities were $39,000. His drawings for the year amounted to $6,000. His...

Discrete Structures Recall C(n,r) counts the number of ways to choose r items from a set of n, without repetition. The formula is C(n,r) = n! / ((n-r)! * r!). a) Using algebra, show that C(n+1,r+1) -...

Answer is complete but not entirely correct. Cash$ 3 1 , 9 0 0 selected answer correctEquity$ 3 8 6 , 9 0 0 selected answer correctFixed assets 3 5 5 , 0 0 0 selected answer correctTotal$ 1 7 , 4 0 0...

What is the between-subjects design?

7. Let a, b, c, D, E be real numbers with c =I- O. (a) If DE > 0, find all extrema of ax + by + cz subject to the constraint z = Dx2 + Ey2. Prove that a maximum occurs when cD

6. Let V be an open set in Rn, a E V, and f : V ---+ R be C2 on V. If f (a) is a local minimum of f, prove that D(2) f (a) (h) 2: 0 for all h ERn.

5. Let V be open in R 2, (a, b) E V, and f : V ---+ R have second-order partial derivatives on V with fx (a, b) = fy (a, b) = O. If the second-order partial derivatives of f are continuous at (a, b)...

Ida Sidha Karya Company is a family-owned company located on the island of Bali in Indonesia. The company produces a handcrafted Balinese musical instrument called a gamelan that is similar to a...

Moving to another question will save this response Quin 1921 Question 19 3 points "A company is selling Toyota cars. An Islamic bank invests USD 10 million for a 50% profit share. The investor...

Martin is offered an investment where for $5,000 today, he will receive $5,250 in one year. He decides to borrow $5,000 from the bank to make this investment. What is the maximum interest rate the...