Question HELP!

The function f is defined for x ≥ 0. It is given that f has a minimum value when x = 2 and that f ′′(x) = (4x + 1)^-1/2.
(i) Find f ′(x).
It is now given that f ′′(0), f'(0) and f(0) are the first three terms respectively of an arithmetic progression.
(ii) Find the value of f(0).
(iii) Find f(x), and hence find the minimum value of f.
Sol. (i) I got f'(x) by intergrating and finding the value of c and then substiuting the value of c to form the equation.
y= (4x+1)^1/2-3)/2
I am really confused on the second and third part. Please help.

(ii)

For any AP, we have

a0 = a
a1 = a+d
a2 = a+2d

So, to start, can you work out a simple relationship between a0,a1,a2, eliminating a and d.

Once you've got that eqiatopm, f'(0) and f''(0) are striaght forward, f(0) is unknown, and those are the three values to go into your AP to find f(0).

I am still confused. Can you make it more clear. Thank you.

Which bit?

From second part.

Of what? My answer, or the original question? Please be clear, and say what bit you'd don't understand, otherwise this is going to take hours.

The original question with your answer as well. Thank you.

The three terms, f ′′(0), f'(0) and f(0) are the first 3 terms of an AP.

You can work out the first two, simply by substituting x=0, into the appropriate formula. f''(x), or f'(x).

To find the third, f(0), we need to establish a relationship between all three, so we can form an equation, with the unknown f(0), and the two other known values.

These three terms are the a0, a1, a2 of the AP.

And follow my first post.