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# Functions, differentiation watch

1. The function f, defined for XER and x>0, is such that

f ' (x) =x^"-2 + 1/x^2 yes that's f dash, i.e. gradient function or dy/dx

(a) Find the value of f ''(4)

(b) Given that f(3)=0, find f(x)

(c) Prove that f is an increasing function

any help is massively appreciated!
2. (a) Differentiate f'(x) and substitute x=4.
(b) Integrate f'(x) remembering the constant of integration, then use f(3)=0 to find the required value of the constant.
(c) You need to show f'(x) >= 0 for all x.
3. f ' (x) =x^"-2 + 1/x^2

a) f'(x) = 1/x^(2) + 1/x^2
= 1/4^(2) +1/(4)^2
= 1/16+1/16
= 2/16

B) f(x) = integral of x^(-2) + x^(-2)
= integral of 2x^(-2)
= -2x^(-1) + C
f(3) = 0
-2(3)^(-1) + C = 0
C = 2(3)^(-1)
C = 1/3
F(x) = -2x^(-1) + (1/3)
4. Sorry
No,

f ' (x) = x^2 -2 + 1/x^2

in words: x sqaured, then minus two, then add 1 over x squared.
you can write that as f'(x) = x^2 + x^-2 -2

differentiate that again and substitute 4 into it to get part a.
Differenciate again??...he allready has F'(x) for part A. He doesn't need to do anything to it until part B where he integrates.
6. Part (c) is a little tricky. See if you can use the fact that

(x^2-1)^2 >= 0 for all values of x
7. OHRIGHT sorry my mistake, didn't realise he was after F''(x)
8. Done parts (a) and (b) now, just need help with proving that f is an increasing function, i.e. the gradient is always >0

Help!!
9. (Original post by Mush)
Differenciate again??...he allready has F'(x) for part A. He doesn't need to do anything to it until part B where he integrates.
Part (a) requires the second derivitive.
10. (Original post by Lawbutwhere?)
Done parts (a) and (b) now, just need help with proving that f is an increasing function, i.e. the gradient is always >0

Help!!
See my post above.
11. dont really see how tht helps, ive got to show that

x^2 -2 + 1/x^2 is always greater than zero.

Sorry, probs just me...lol
12. Multiply it out, see what you get.
13. (Original post by Lawbutwhere?)
Done parts (a) and (b) now, just need help with proving that f is an increasing function, i.e. the gradient is always >0

Help!!
increasing functionf(x) f'(x)>0 .
x²-2+1/x²>0
(x-1/x)²>0
True for all x except f'(1)=0 .:. increasing function

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