Question

How do you show that the curve `6x^3+5x-3` has no tangent line with slope 4?

Answer 1

Find the derivative. Set it equal to `4`. Explain why there is no (real) solution.

Explanation:

For `y=6x^3+5x-3`, the slope of the tangent line is given by

`y' = 18x^2+5`.

In order to have a tangent line with sloe `4` there would have to be an `x` such that

`18x^2+5=4`.

But that is true only if `x^2 = -1/18`.

For real numbers `x`, we cannot have `x^2` is a negative number.