At what points do the functions #y = x^2x# and #y = sin pix# intersect?
1 Answer
These two equations intersect at the points
Explanation:
First, let us take a look at:
#y = x^2x#
We can factor this as:
#y = x(x1)#
so this quadratic has
It has minimum value at the midpoint of these two
#y = color(blue)(1/2)(color(blue)(1/2)1) = 1/4#
So note that
The intersections of
#0 = x^2x1#
#color(white)(0) = x^2x+1/45/4#
#color(white)(0) = (x1/2)^2(sqrt(5)/2)^2#
#color(white)(0) = (x1/2sqrt(5)/2)(x1/2+sqrt(5)/2)#
That is:
#x = 1/2+sqrt(5)/2#
Note that
Note that
So
Outside these intervals,
Now consider

If
#x in [1/2sqrt(2), 0)# then#sin(pix) < 0# 
If
#x=0# then#sin(pix) = 0 = x^2x# 
If
#x in (0, 1)# then#sin(pix) > 0# 
If
#x = 1# then#sin(pix) = 0 = x^2x# 
If
#x in (1, 1/2+sqrt(5)/2)# then#sin(pix) < 0#
So in each of the intervals
So the only two points of intersection are:
#(x, y) = (0, 0)#
#(x, y) = (1, 0)#
graph{(yx^2+x)(y  sin(pix)) = 0 [2.105, 2.895, 1.19, 1.31]}