Question

How do you find the points where the graph of the function # f(x) = 1+sinxcosx# has horizontal tangents and what is the equation?

Answer 1

`x = pi/4 + pin` and `(3pi)/4 + pin`

Explanation:

Differentiate.

`f'(x) = cosx(cosx) + sinx(-sinx)`

`f'(x) = cos^2x- sin^2x`

`f'(x) = cos2x`

The tangent will be horizontal when it has a slope of `0`. Set the derivative to `0` and solve:

`0 = cos2x`

`2x = pi/2 and (3pi)/2`

`x = pi/4 and (3pi)/4`

If you add in periodicity, you have:

`x = pi/4 + pin` and `(3pi)/4 + pin`

Because `cos2x` has a period of `pi`.

Hopefully this helps!