after doing two steps in the cutting technique
i see that the solution lies [1.75,2]
which number to put in the referson formula to find the solution
show that has a root on [1,2]
use the two steps of the crossing method that contains the root.
find the root of the equation using newton referson methos.
solution of the book:
so it rises and the is no root in [0,pi/3] interval
then they say that in [pi/3,pi] f(x) going down because f'(x)<0
so there is a solution in this interval
but how does f'(x)<0 shows that there is a soltion??
in order to see that there is a solution we need to have two point one positive one negative and that f(x) is going down
why they check from 0 to pi
and not from [1,2] like asked in the question
Then and and as is continuous and changes sign on it has a root in
Now , so we replace the bottom limit to get a new interval for our root.
Next step the new trial point is and , so again we replace the bottom limit to get the new interval .