Prove that there is no line tangent to the curve `y=4x^2` that contains the point (1,5):
The slope of a tangent line to `y=4x^2` at x is m=8x.
The slope between the points (1,5) and a point on the graph `(x,4x^2)` is `m=(4x^2-5)/(x-1)` .
The discriminant is `(-8)^2-4(4)(5)=-16` so there are no real solutions. Thus there is no value for x such that the tangent line to the given curve includes the point (1,5).
(The pencil of tangent lines to a parabola all lie "outside" the curve, so no point in the "interior" will be included in any of the tangent lines. )