Question: Prove that, if f'(x) exists and is continuous on an interval l and if f'(x) 0 at all interior points of l, then either

Prove that, if f'(x) exists and is continuous on an interval l and if f'(x) ≠ 0 at all interior points of l, then either f is increasing throughout l or decreasing throughout l.

Step by Step Solution

★★★★★

3.34 Rating (160 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Suppose that there are points x 1 and x 2 in I where fx 1 0 and fx 2 0 ... 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)

955-M-C-D-E (1788).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

Let f be a functional on an open subset S of n. Then f is continuously differentiable (C1) if and only if each of the partial derivatives Dif[x] exists and is continuous on S. We now present some...

Suppose that f' exists and is continuous on a nonempty, open interval (a, b) with f(x) 0 for all x (a, b). a) Prove that f is 1-1 on (a, b) and takes (a, b) onto some open interval (c, d). b) Show...

Prove that if |f(x)| Then lim g(x) = 0. lim f(x)g(x) = 0.

The colored writing below are the notes my professor is asking the work to be done like. I'm unsure as he didn't like the way my work is done. ex: ) lim As X-8 , X-8- (x-8) M F(x)- Note: Let Fix) =...

Test Review 4 Math 1241 Name:_______________________________________ Chapters/Sections 3.7 - 4.7 Part I - Multiple Choice - You must show ALL your work to receive credit For #1-5, Use L'Hospital's...

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...

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...

NO NEED FOR A EXPLANATION, JUST ANSWER ONLY. Q1. Question 1 of 5 - / 10 .. . Current Attempt in Progress A function f has a removable discontinuity at x = cif lim f (x) exists by f is not continuous...

Theorem 1 If lim f x L and lim f x M then L M x p x p Proof Suppose LM then I L MI 0 Since lim f x L if we take x p then 38 0 such that Ix pl 8 1 f x LI e Similarly since lim f x M 38 0 such that x p...

Theorem 1 If lim f x L and lim f x M then L M x p x p IL MI Proof Suppose L M then IL MI 0 Since lim f x L if we take 2 then 38 0 such that x p Ix pl 8 f x LI E Similarly since lim f x M 38 0 such...

In Crane Corporation's first year of operations, inadequate budgeting led to significant cost overruns in multiple production areas. Direct materials needed to be express-delivered, and laborers...

Write a python program that gets the user's name, and prints its last 2 characters.

CHALLENGE Fuel efficiency t test. Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per...

In August Victoria Air Conditioning, Inc. (VAC), entered into a subcontract for insulation services with Southwest Texas Mechanical Insulation Company (SWT), a partner ship composed of Charlie Jupe...

Consider two individuals who take a very different view of life and consider how this shapes their tastes over intertemporal tradeoffs. A: Jim is a 25 year-old athlete who derives most of his...

Consider my tastes for paperclips and all other goods (denominated in dollar units). A: Suppose that my willingness to trade paper clips for other goods does not depend on how many other goods I am...

I love the Austrian candy Mozartkugeln. They are a small part of my budget, and the only factor determining my willingness to pay for additional Mozartkugeln is how many I already have. A: Suppose...

Trinance is an online trading platform offering trading services to investors in cryptocurrencies. The company started off as a small private company and has grown strongly over the past fifteen...

Kurt owns a convertible bond that matures in three years. The bond has an 8 percent coupon and pays interest semi-annually. The face value of the bond is $1,000 and the conversion price is $25....

The owner of an antique store estimates that there is a 35% chance she will make $2400 when she sells an antique chinacabinet, a 60% chance she will make $900 when she sells thecabinet, and a 5%...