Question: 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)

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 ˆˆ Rn{0}.
c) If T := -Df(a)/f2(a), show that

for ||h|| sufficiently small.
d) Prove that 1/f(x) is differentiable at x = a and

e) Prove that if f and g are real-valued vector functions which are differentiable at some a, and if g(a) ‰ 0, then

f(a + h)-f(a) _ T(h) = f(a)f (a + h) (fat. h)-f(a )) Df(a)(h) f3(a)/(a + h) D() (a)=-Dra) (a) =-T2(a) g(a) Df(a)-f(a) Dg(a) g2(a) a)

Step by Step Solution

★★★★★

3.50 Rating (160 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a By modifying the proof of Lemma 328 we can prove that if fa 0 t... 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 (640).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Suppose that z = F(x, y) is differentiable at (a, b), that Fy(a, b) 0, and that I is an open interval containing a. Prove that if f: R is differentiable at a, f(a) = b, and F(x, f(x)) = 0 for all x...

Let r > 0, let a Rn, and suppose that g: Br(a) Rm is differentiable at a. a) If f: Br(g(a)) R is differentiable at g(a), prove that the partial derivatives of h = f o g are given by for j = 1,...

Suppose that f: R R is differentiable and periodic and that f' is integrable on [-Ï, Ï]. Prove that a(f') = kb;(f) and b(f)= -kar(f), ke N.

4. [QUOTIENT RULE] Let f : Rn -> R be differentiable at a with f (a) f 0. (a) Show that for Ilhll sufficiently small, f(a+h) f 0. (b) Prove that D f(a)(h)/llhll is bounded for all h ERn \ {o}. (c)...

Exercise 4.3.7. [Used in Lemma 7.3.4 and Theorem 7.3.12.] Let [a, b] C R and [b, c] C R be non-degenerate closed bounded intervals, and let f: [a, b] - R and g: [b, c] - R be functions. Suppose that...

(a) Let F: D + RN for some open subset D CRN and let the standard assumptions hold, then there exists some 8 >0 such that given x) e B:(x*), the inexact Newton iteration x(x+1) = x() - ] (x(K)-F...

Please answer questions a to e according to the following formula, and give detailed steps and explanations Recall that for a function f : RN - R, the gradient is simply the N x 1 vector formed by...

Hello. Kindly solve these 5. Consider a market for used cars, where there are 300 lemons and 600 peaches. Buyers value peaches at $12,000 and lemons at $6,000. (a) If all cars (lemons and peaches)...

1/2 12 1. Let VCR be a subset of Rn. We say that V is measure-0 if there exists a sequence of sets U1, U2, . such that each U; is a union of cubes [a1, b] .. [an, bn] which only overlap on their...

10. Let r > 0, (a, b) E R 2 , f : Br (a, b) -+ R, and suppose that the first-order partial derivatives fx and fy exist in Br (a, b) and are differentiable at (a, b). (a) Set ~(h) = f(a + h, b + h) -...

Shop-Aid Tools is a small manufacturer of home workshop power tools. The company employs 30 production workers and 10 administrative persons. The following procedures are used to process the...

Which of the following accounts is not included in the calculation of net income? 0 1 : 2 0 : 3 7 Multiple Choice Cash. Rent expense. Rent revenue. Wages expense. Services revenue.

Listening to music has long been thought to enhance intelligence, especially during infancy and childhood. To test whether this is true, a researcher records the number of hours that eight...

4. Let r> 0, I: Br(O) --t R, and suppose that there exists an a > 1 such that II(z)1 ::; Ilzll'" for all z E Br(O). (a) Prove that 1 is differentiable at O. (b) What happens to this result when a = 1?

3. Prove that the following function is not differentiable at (0,0).

2. Prove that I(x, y) = JiXYI is not differentiable at (0,0).

Assume the following characteristics for a particular bond: Face value=$1000 Annual coupon payment=$60 (First payment due in 1 year) Internal yield to maturity = 7% Term= 3 years 1. Compute the...

B H M sume you want to receive $5,000 at the end of each month after retirement. If you can earn 3.8% per month and you expect to need the income for 25 years, how much you need to have in your...

The risk-free rate of return is 4.5%, the expected rate of return on the market portfolio is 14%, and the stock of Xyrong Corporation has a beta coefficient of 3. Xyrong pays out 40% of its earnings...