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

Oct 25, 2015

See the explanation section.

Explanation:

$f \left(x\right) = \sin x - \cos x$

$f ' \left(x\right) = \cos x + \sin x$

$f ' ' \left(x\right) = - \sin x + \cos x$

$f ' ' \left(x\right) = 0$ where $\sin x = \cos x$ or $\tan x = 1$

This happens at $x = \frac{\pi}{4} + \pi k$ for integer $k$.

For $\frac{\pi}{4} < x < \frac{5 \pi}{4}$ we have $\sin x > \cos x$ so $f ' ' \left(x\right) < 0$ and the graph of $f$ is concave down.

For $\frac{- 5 \pi}{4} < x < \frac{\pi}{4}$ we have $\sin x < \cos x$ so $f ' ' \left(x\right) > 0$ and the graph of $f$ is concave up.

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

on intervals of the form $\left(\frac{\pi}{4} + 2 \pi k , \frac{5 \pi}{4} + 2 \pi k\right)$, where $k$ is an integer, the graph of $f$ is concave down.

on intervals of the form $\left(\frac{- 5 \pi}{4} + 2 \pi k , \frac{\pi}{4} + 2 \pi k\right)$, where $k$ is an integer, the graph of $f$ is concave up.

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