Consider the function f(x)=sin(2x)-1 and g(x)= 2 cos^2(x), both with domain [0,phi]. the line x = p intersects the graph of f in point a and the graph of g in point b. calculate the value of p for which the length of segment ab is maximal. you kindly answer this question

I think the correct value of p is 2.9 when dealing with this question.  Since AB is a vertical line between f(x) and g(x), you know you subtract the upper function by the lower function to find the distance between them.  In other words you are tying to find the x value (which can also be thought of as p) that produces the greatest value when g(x) is subtracted from f(x). The reason why g(x) is subtracted by f(x) is that g(x) is above f(x) over the domain of 0 to pi. 
I found p by making the function d(x)=g(x)-f(x).  Then I took the derivative of d(x) and found when the derivative of d(x) equals 0 since that indicates a maximum or a minimum of the function d(x).  their ended up being 2 points from 0 to pi where the derivative of d(x) equaled 0.  one at 2.9 and the other at 1.34.  The value at 1.34 is a minimum and therefore can't be the right answer.  The value at 2.9 is a maximum and is therefore the correct answer.
By the way I did all of my work on a graphing calculator which makes these types of questions a lot easier since it makes it so that you don't have to do it by hand. 
I hope this helps. Let me know if I got anything wrong in the comments so that I can try to fix it.