C2 Differentiation Question

Hi,

I'm stuck on this differencitation question on increasing functions and hope that someone can help.

The function f, defined for x E R (symbols that look like E and R) , x>0 is such that

f'(x) = x²-2+1/x²

Prove that f is an increasing function.

In the back of the book it says " f'(x) = (x-1/x)² etc ". So I thought I'd use this and say anything squared is greater than or equal to 0, but then dy/dx would have to be greater than 0 if this was an increasing function, and it could equal 0 if x=1. So I'm not sure how to do this question now. Any help appreciated.

Thanks! :smile:

Reply 1

steve2005

17

sweet_gurl

Hi,

I'm stuck on this differencitation question on increasing functions and hope that someone can help.

The function f, defined for x E R (symbols that look like E and R) , x>0 is such that

f'(x) = x²-2+1/x²

Prove that f is an increasing function.

In the back of the book it says " f'(x) = (x-1/x)² etc ". So I thought I'd use this and say anything squared is greater than or equal to 0, but then dy/dx would have to be greater than 0 if this was an increasing function, and it could equal 0 if x=1. So I'm not sure how to do this question now. Any help appreciated.

Thanks! :smile:

For an increasing function f'(x)>0
Therefore x²-2+x-²>0
x^4 - 2x^2+1>0
(x^2 - 1)(x^2 - 1)>0
At critical points x=1
The curve is valley shaped. We are told x>0 so fuction is increasing for x>1

Reply 2

Malik

12

sweet_gurl

Hi,

I'm stuck on this differencitation question on increasing functions and hope that someone can help.

The function f, defined for x E R (symbols that look like E and R) , x>0 is such that

f'(x) = x²-2+1/x²

Prove that f is an increasing function.

In the back of the book it says " f'(x) = (x-1/x)² etc ". So I thought I'd use this and say anything squared is greater than or equal to 0, but then dy/dx would have to be greater than 0 if this was an increasing function, and it could equal 0 if x=1. So I'm not sure how to do this question now. Any help appreciated.

Thanks! :smile:

You've basically done it you finish off by saying "this is true for all accept when x=1" or something similar for the full marks.

Reply 3

sweet_gurl

OP

4

steve2005

For an increasing function f'(x)>0
Therefore x²-2+x-²>0
x^4 - 2x^2+1>0
(x^2 - 1)(x^2 - 1)>0
At critical points x=1
The curve is valley shaped. We are told x>0 so fuction is increasing for x>1

Thanks. Are critical points the same as turning points?

Reply 4

Dekota

sweet_gurl

Thanks. Are critical points the same as turning points?

Nope.

Reply 5

steve2005

17

sweet_gurl

Thanks. Are critical points the same as turning points?

We have a quadratic and IF it equaled zero we could equate each bracket to zero and say x= whatever.

I use the term critical point as a way of giving the value of x at a particular point. The solution then depends on the shape of the graph.

--------------

sweet_gurl

Thanks. Are critical points the same as turning points?