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

1 Answer
Oct 25, 2015

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.