Here is the question: the diagram(which I am unable to draw) has two parabolas in the first quadrant, one with a minimum at (3,7) and one with a maximum at (2,4) It has a line which touches the top parabola at A and the bottom parabola at B the expression for the top parabola is y=(x-3)^2 +7 where the line joins it at A and the expression for the bottom parabola is y=x(4-x)where the line joins it at point B Point A and point B have the same x coordinate. (which means that it is perpendicular to the x axis and the expression for the line is x=? with no y value given. Find an expression for the length of AB and determine its minimum length. The length of the line AB must be an expression of the y terms which is the height of the line. A is at y=7+A and B is at y= 4-B so an expression for the length of the line is y= A+B+3 taking into account the 3y units between the maximum and the minimum of the two parabolas. What I don't understand is how to use differentiation to find the minimum length but I assume it depends on finding the value for x for the line