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Prove f(x) is an increasing function

(c)
f`(x) > 0 needed, or f`(x) ≥ 0, or “as x increases, f(x) increases”

f`(x) = (x − 1/x)^2, > 0 always, or ≥ 0 always

Is the answer to the following question:

f ′(x) = x^2 – 2 + x^(-2).

c) Prove that f is an increasing function.



I understand the first line of the answer, that the gradient has to be > 0, but I don't understand the proof part (line 2). Could someone explain how to prove it please?
You need to prove that f`(x) > 0 for all real x.
i think you find dy/dx, factorise it, find the values for x, substitute them into dy/dx, if answer >0 = increasing function.

orr you substitute a few values for x into dy/dx ( example, a negative one and a posotive), they should be more than 0.
Reply 3
Thanks for the replies, but I still don't understand how to prove it is an increasing function.
For it to be an increasing function, f'(x) must always be greater than or equal to 0, for any value of x.

If you have f'(x) = x^2 – 2 + x^(-2), you need to show this is always greater than or equal to 0, to show it is always increasing. A good way to do this is to spot that it is a perfect square i.e. f'(x) = (x − 1/x)^2. Remember that squares are always positive (or 0).
Reply 5
ok.. you know if a function is increasing it's derivative is positive, yeh?

so look at the derivative...

can you factorise it to show that it is always positive?

think about f'(x) = x^2 - 2x + 1 for example

we can write this as (x-1)^2

which is always greater than zero.. so f'(x) is always positive and so f(x) is always increasing

understand?
Reply 6
By using these perfect squares, you are not proving for all x that are real numbers. You are proving for those numbers that you made an exampe with. What if there is a counter example that you knew and avoided making an example with? Please prove for all x that are real numbers
Original post by Waheyyyy
(c)
f`(x) > 0 needed, or f`(x) 0, or “as x increases, f(x) increases”

f`(x) = (x 1/x)^2, > 0 always, or 0 always

Is the answer to the following question:

f ′(x) = x^2 2 + x^(-2).

c) Prove that f is an increasing function.



I understand the first line of the answer, that the gradient has to be > 0, but I don't understand the proof part (line 2). Could someone explain how to prove it please?


You need to prove the derivative is always greater than 0, therefore the gradient is always positive and the function is increasing at all points.

Posted from TSR Mobile
Original post by majmuh24
You need to prove the derivative is always greater than 0, therefore the gradient is always positive and the function is increasing at all points.

Posted from TSR Mobile


This question is from 2009 :tongue:
Original post by SecretDuck
This question is from 2009 :tongue:


-_-

Someone else bumped it before me, not my fault :tongue:

Posted from TSR Mobile
Original post by majmuh24
-_-

Someone else bumped it before me, not my fault :tongue:

Posted from TSR Mobile


Sure :wink2:
Original post by SecretDuck
Sure :wink2:


No arguing :spank:

I've had my fair share of necroing study help threads
what would i do if i can't factorise the function f'(x)=3x^2-12x+17
Original post by Lekokoto
By using these perfect squares, you are not proving for all x that are real numbers. You are proving for those numbers that you made an exampe with. What if there is a counter example that you knew and avoided making an example with? Please prove for all x that are real numbers


What are you talking about?


Posted from TSR Mobile
Original post by priyaroy789
what would i do if i can't factorise the function f'(x)=3x^2-12x+17


Complete the square


Posted from TSR Mobile
Original post by physicsmaths
Complete the square


Posted from TSR Mobile



Can you complete the square if there are numerous x terms?
Original post by Engineerrookie
Can you complete the square if there are numerous x terms?


Don't know what you mean.

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