recall that the derivative gives the slope of the tangent line to a curve at some point.
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)