Find the coordinates of the turning point on the curve \(y=x^2-2bx+c\). Sketch the curve in the case that the equation \(x^2-2bx+c=0\) has two distinct real roots. Use your sketch to determine necessary and sufficient conditions on \(b\) and \(c\) for the equation \(x^2-2bx+c=0\) to have two distinct real roots. Determine necessary and sufficient conditions on \(b\) and \(c\) for this equation to have two distinct positive roots.
Find the coordinates of the turning points on the curve \(y=x^3-3b^2x+c\) (with \(b>0\)) and hence determine necessary and sufficient conditions on \(b\) and \(c\) for the equation \(x^3-3b^2x+c=0\) to have three distinct real roots. Determine necessary and sufficient conditions on \(a\), \(b\) and \(c\) for the equation \((x-a)^3-3b^2(x-a)+c=0\) to have three distinct positive roots.
Show that the equation \(2x^3-9x^2+7x-1=0\) has three distinct positive roots.
