Question

Sketch the parabolas `y=x^2` and `y=x^2-2x+2`, do you think there is a line that is tangent to both curves?

Answer 1

No there is not

Explanation:

This is `y=x^2`
graph{x^2 [-10, 10, -2,10]}

This is `y=x^2-2x+2`
graph{x^2-2x+2 [-10, 10, -2, 10]}

This is both:
graph{(y-x^2+2x-2 )(y-x^2)=0[-10, 10, -2, 10]}

The gradient of the tangent at any particular point is given by the value of the derivative at that point.

For;

`y=x^2 => y'=2x`
`y=x^2-2x+2 => y'=2x-2`

If we were to have a common tangent we would require that derivatives have a common solution (NB this is a necessary but not sufficient condition). i.e.

`2x = 2x-2`

Which clearly has no solution `=>` there is no common tangent.