# c2 differentiation - increasing functionWatch

Announcements
#1
f '(x)=x² - 2 + 1/x²

so for increasing function, f '(x) > 0

⇒ x² - 2 + 1/x² > 0

so my question is how do i go about from here to show this is true for all real values of x.

the answer provided is the equation simplified to (x - 1/x)² > 0

my other question would be how did they simplify that eqn?

thanks
0
13 years ago
#2
x² - 2 + 1/x² = x2 - x/x - x/x + 1/x2 = (x - 1/x)(x - 1/x) = (x - 1/x)2
0
13 years ago
#3
So you want to prove f(x) is an increasing function?

The condition is that the derivative of f(x) is greater than zero for all x.

It provides you with the expression for the derivative: x2 - 2 + 1/x2

This can be FACTORISED to (x - 1/x)2. If it's not obvious why, look at the following:

x2 - 2 + 1/x2

Take out 1/x2 as a factor:

x-2(x4 - 2x2 + 1)

Let y = x2

y-1(y2 - 2y + 1)

There's a familar quadratic in the brackets, which you know how to factorise...

y-1(y-1)2

Then put back in y = x2...

x-2(x2 - 1)2

Now, x-2 = (x-1)2, so you can take it inside the bracket as x-1...

(x - x-1)2

Which is equivalent to (x - 1/x)2.

It's just a simple factorisation. If you expand that bracket, you'll get the original expression.

The final step of the argument is to say that since the derivative is expressed as a perfect square, then there is no value of x which will make it negative. Also, since there is a 1/x term, x must not be zero!

Hence, the derivative must always be greater than zero.
0
#4
thank you worzo for the clear explanation. +ve rep has been sent!!
0
X

new posts
Back
to top
Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

### University open days

• Norwich University of the Arts
Thu, 23 Jan '20
• SOAS University of London
Development Studies, Interdisciplinary Studies, Anthropology and Sociology, Languages, Cultures and Linguistics, Arts, Economics, Law, History, Religions and Philosophies, Politics and International Studies, Finance and Management, East Asian Languages & Cultures Postgraduate
Sat, 25 Jan '20
• University of Huddersfield
Sat, 25 Jan '20

### Poll

Join the discussion

#### - Have you considered distance learning for any of your qualifications?

Yes! I'm on a distance learning course right now (9)
7.96%
Yes, I've thought about it but haven't signed up yet (12)
10.62%
No, but maybe I will look into it (28)
24.78%
No and I wouldn't consider it (64)
56.64%