Question

How do you determine whether the function `f(x)= sinx-cosx` is concave up or concave down and its intervals?

Answer 1

See the explanation section.

Explanation:

`f(x)= sinx-cosx`

`f'(x)= cosx+sinx`

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

`f''(x) = 0` where `sinx = cos x` or `tanx=1`

This happens at `x=pi/4 + pik` for integer `k`.

For `pi/4 < x < (5pi)/4` we have `sinx > cos x` so `f''(x) <0` and the graph of `f` is concave down.

For `(-5pi)/4 < x < pi/4` we have `sinx < cos x` so `f''(x) > 0` and the graph of `f` is concave up.

Both sine and cosine are periodic with period `2pi`, so

on intervals of the form `(pi/4+2pik, (5pi)/4+2pik)`, where `k` is an integer, the graph of `f` is concave down.

on intervals of the form `((-5pi)/4+2pik, pi/4+2pik)`, where `k` is an integer, the graph of `f` is concave up.

There are, of course other ways to write the intervals.