The points and are on the curve such that is a right angle. What points and will give the smallest possible area for the triangle ?
Writing the coordinates of the points and as and gives the line gradient .
The line has gradient: .
Solving gives and hence for the coordinates of .
By dropping perpendiculars to the x-axis, and labelling these points and , the area of the triangle can be found by subtracting the area of two triangles from a trapezium:
Solving with calculus:
when (or ). This gives and at (1,1) and (-1,1).
Solving without calculus:
which has a minimum value when i.e. when (or ).
This gives and at (1,1) and (-1,1).