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)` .

`(4x^2-5)/(x-1)=8x`

`4x^2-5=8x^2-8x`

`4x^2-8x+5=0`

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 graph:

(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. )

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