is a function defined at such that , and when :
Let
Find the sum of x-coordinates of all the intersection points of and in the interval .
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
f ( 1 + x ) = f ( 1 − x ) , which means whatever value f attains on moving x distance from 1 to the right is the same as moving x distance from 1 to the left , thus function f ( x ) is symmetric about the line x = 1 and simillary g ( x ) is also symmetric about x = 1 , as g ( 1 + h ) = ln ( 1 + h ) and g ( 1 − h ) = ln ( 2 − 1 + h ) = ln ( 1 + h ) , where h is any positive real number. This symmetry of f and g about x = 1 line will help us in evaluating the sum of x -coordinates.
We are looking for intersections in the interval [ 1 − 6 , 1 + 6 ] . If f and g intersect at x = 1 + a then they will also intersect at x = 1 − a due to symmetry. Thus the sum of x -coordinates will be 1 + 2 × (No. of times the graph intersect in ( 1 , 7 ] ). By making the graph, we observe that they intersect 3 times in the interval ( 1 , 7 ] . The answer will be 7 . An alternative method to calculate the answer will be appreaciated !