Use Newton's method to find the coordinates of the inflection point of the curve y = ecos(x),
0 ≤ x ≤ π , correct to six decimal places.
Do you mean $\displaystyle y= e^{cos x}$?
As Captain Black said, an inflection point occurs at a local extremum of y' which, in turn, mean that y"= 0 (since y" always exists for this function).
Take the second derivative of this function, set it equal to 0 and use Newton's method to solve the equation.