Question

find the point on the curve y= cosx closest to the point (0,0)

Answer 1

Distance formula:
d = SQRT( (x1-x2)^2 + (y1-y2)^2 )
Here, x1 = y1 = 0.
d = SQRT( x^2 + y^2 )
y = cos(x) - equation of our curve
d = SQRT( x^2 + (cos(x))^2 )
f(x) = x^2 + (cos(x))^2

f'(x) = 2x - 2cos(x)sin(x) = 2x - sin(2x)
Set f'(x) to zero:
2x - sin(2x) = 0
2x = sin(2x)
2x = 0, or x=0 - the unique solution that minimize the distance
cos(0) = 1

So, the point on the curve y=cosx closest to the point (0,0) is point (0,1).