Question

How do you find the points where the graph of the function `f(x) = x^4-4x+5` has horizontal tangents and what is the equation?

Answer 1

at `(1,2)`; equation of `y=2`

Explanation:

A horizontal tangent occurs whenever the function's derivative equals `0`, since a value of `0` represents that the function's tangent line has a slope of `0`. Lines with slope `0` are horizontal.

To find the function's derivative, use the power rule.

`f(x)=x^4-4x+5`

`f'(x)=4x^3-4`

Find the points when `f'(x)=0`.

`4x^3-4=0`

`4x^3=4`

`x^3=1`

`x=1`

There is a horizontal tangent at `(1,2)`, thus its equation is `y=2`.

We can check a graph of `f(x)`:

graph{(x^4-4x+5-y)(y-0x-2)=0 [-19.92, 20.63, -3.52, 16.74]}