now, the derivative here is we need to show that there are three such lines with this slope touching our curve.
now, any point on our curve is given by:
thus, using , the slope of any line touching our curve and passing through the point (2,3) is given by:
now, if this slope is equal to the slope of our tangent line, we will have a tangent for every x-value given by that formula. thus we want to solve for all x's such that:
i leave the rest to you. showing that we have 3 solutions to this equation shows that there will be 3 tangent lines to our curve that pass through (2,3)