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

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

To determine whether the function $f(x) = \sin(x) - \cos(x)$ is concave up or concave down and its intervals, you need to find the second derivative of the function and then examine its sign.

1. Find the first derivative: $f'(x) = \frac{d}{dx} (\sin(x) - \cos(x)) = \cos(x) + \sin(x)$

2. Find the second derivative: $f''(x) = \frac{d^2}{dx^2} (\cos(x) + \sin(x)) = -\sin(x) + \cos(x)$

3. Determine the intervals where $f''(x) > 0$ for concave up and where $f''(x) < 0$ for concave down.

4. Solve $-\sin(x) + \cos(x) > 0$ for concave up: $-\sin(x) + \cos(x) > 0$ $\cos(x) > \sin(x)$

5. Analyze the sign of $\cos(x) - \sin(x)$ within the period $0 \leq x < 2\pi$ or $0^\circ \leq x < 360^\circ$ to determine the intervals where the function is concave up.

6. Repeat the process to determine the intervals where the function is concave down by analyzing where $f''(x) < 0$.

By following these steps, you can identify the intervals where the function $f(x) = \sin(x) - \cos(x)$ is concave up or concave down.