C2 question. Little help? :)

Thank you for the heads up!

https://ibb.co/nmGfkS

Hi, thank you for re-uploading.

There are several ways to approach this question:

One way is by rewriting it as: x^2 - 2 + x^-2 if that helps.

(x-1/x)(x-1/x) or (x-1/x)^2 is factorising it. Straight away you can say (x-1/x)^2 >= 0 for all values of x because squaring always gives a positive result. However, for x > 0, (x - 1/x) > 0 for all values of x, since 1/x is < x for x > 0. Therefore, (x-1/x) > 0 for x > 0 and therefore squaring it will give you a result > 0, therefore it is an increasing function.

If you want to prove something is an increasing function, you prove it's derivative is > 0 for all values.
Usually, I'd show that e.g. x^2 >=0 for x > 0 and x^-2 > 0 for x > 0.
However, that doesn't show it's positive for x > 0 because of the -2.

So, the way I'd do it is actually rewrite f'(x) as: x^2(x^2 - 2 + x^-2)/x^2 = (x^4 - 2x^2 + 1) / x^2 = (x^2 - 1)^2/x^2
The numerator is always positive because it is squared and we know x^2 is always positive. Both functions are >0 for x > 0, so they are definitely increasing functions.

Can't f'(x) also be 0 when x=1? How can it be an increasing function if it isn't always positive?

Yeah you guys are legends!!! Thank you. I wrote out your notes on paper for reference. The x^2/x^2 is a lovely touch. That cleared things up for me. The moment there's a reciprocal or anything slightly out of the box to factorise I come a bit unstuck. I really appreciate your time and patience. Thank you.