Question: Suppose that f is differentiable on (a, b), continuous on [a, b], and that f(a) = f(b) = 0. Prove that if f(c) 0

Suppose that f is differentiable on (a, b), continuous on [a, b], and that f(a) = f(b) = 0. Prove that if f(c) ≠ 0 for some c ∈ (a, b), then there exist x1, x2 ∈ (a, b) such that f'(x1) is positive and fʹ(x2) is negative.

Step by Step Solution

★★★★★

3.08 Rating (159 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

By the Mean Value Theorem fc fc fa c afx 1 ... 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 (278).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

(a) Suppose that f is differentiable on R and has two roots. Show that has at least one root. (b) Suppose f is twice differentiable on R and has three roots. Show that has at least one real root. (c)...

Suppose that f is differentiable on R. If f(0) = 1 and |f(x)|

Suppose that f is differentiable on (0, ). If L = limx f(x) both exist and are finite, prove that L = 0.

Suppose that f is differentiable on [a, b], but the derivative may be discontinuous. (a) Suppose that f ' (a)

7. Let f be differentiable on (a, b), continuous on [a, b], with f (a) = f (b) = O. Prove that if f (c) > 0 for some c E (a, b), then there exist Xl, x2 E (a, b) such that f'(xd > 0 > f'(X2).

1. Suppose that a continuous function f : (a, b) -> R is differentiable except possibly at xo E (a, b), and that lime-+x0 f' (x) = L exists. Prove that f' (To) = L. Warning: It is NOT TRUE in general...

Exercise 3(Taylor formula (42pts)).Assumef: [a, b]?Ris of classCkfork?N. That is to say,fis differentiable on(a,b), its derivativef?is differentiable on(a,b), its second derivativef??is...

10. Suppose that 9 is differentiable on [a, b] and g' is integrable on [a, b]. (a) Prove that if f is continuously differentiable and increasing on [a, b] and 9 is positive on (a, b) with g (b) = g...

please solve #4 parts a - c asap ? 4. (18 points) In part (a), you do not need to justify your answer. a. (2 pts) Give an open cover of the set S = (1,3] that has no finite subcover. b. (8 pts)...

ACORN Timeout x tut8.pdf: MAT 137Y F/W 2020-20 x [ mat137_hw4.pdf: MAT 137Y F/W x () mat137_hw4 (1).pdf * + X O X1+ | C:/Users/User/Downloads/mat137_hw4%20(1).pdf . . 1 12 - + ment is worth 10...

Ronald Fisher published an analysis on the sizes of iris sepals and petals. The data, collected over several years by Edgar Anderson, were used to show that these measurements could be employed to...

Which of the following is considered copyright infringement? A copyrighted poem used without the permission of the author O An image denoted as public domain O Work with expired copyright Facts

During the current year, merchandise is sold for $186,500 cash and $1,437,200 on account. The cost of the goods sold is $900,000. What is the amount of the gross profit? fill in the blank 1 of 1$

(Regulating Natural Monopolies) The following graph represents a natural monopoly. a. Why is this firm considered a natural monopoly? b. If the firm is unregulated, what price and output would...

9. Let 1 : R --; R. Prove that 1 is continuous on R if and only if 1-1 (I) is open in R for every open interval I.

Let Y be a subspace of X. (a) Show that a set V is open in Y if and only if there is an open set U in X such that V = UnY. (b) Show that a set E is closed in Y if and only if there is a closed set A...

7. Show that Theorem 10.40 is best possible in the following sense. (a) There exist sets A, Bin R such that (A U B)O i A~ U BO. (b) There exist sets A, B in R such that A n B =f A n B. (c) There...

Narda Corporation agreed to sell all of its capital stock to Effie Corporation for three monthly payments of $200,000. After Effie made the first required payment, it ceased making other payments....

Hybrid cars are touted as a "green" alternative; however, the financial aspects of hybrid ownership are not as clear. Consider the 2014 Edsel 550h, which had a list price of $5, 900 (including tax...

Which of the following statements is false? A. Redemptions and partial liquidations that qualify as exchanges of the redeemed stock GENERALLY reduce earnings & profits by an amount which does not...