Which of these statements is/are correct? The values are correct but am not sure about the interval. Are both correct, or is only one correct?
f(x)= sinx+cosx , 0 ≤ x ≤ 2π
f(x) is decreasing on interval $$(\frac π4, \frac{5π}{4})$$
f(x) is decreasing on interval $$[\frac π4, \frac{5π}{4}]$$
The derivative of $f(x)$ is : $$f'(x) = \cos(x) - \sin(x)$$
If the derivative is positive, the function is increasing, if it is negative, the function is decreasing. If it is zero at a point, then the point could be a local min, max or an inflection point. Let us find when is the function increasing: $$\cos(x) - \sin(x)>0 \implies \cos(x) > \sin(x)$$ In the interval $0 \le x \le 2\pi$, this is true for $$\color{green}{x \in [0,\pi /4)}$$ and $$\color{green}{x \in (5\pi /4, 2\pi]}$$
When the function is decreasing, the derivative is negative, i.e., $$\cos(x) - \sin(x)<0 \implies \cos(x) < \sin(x)$$This is true for $$\color{green}{x \in (\pi/4, 5\pi /4)}$$
There is local max at $x = \pi/4$ and a local min at $x = 5\pi/4$ (can be found easily using the *first derivative test*). But I'm skipping that part since the question did not ask for that. Here is a graphical representation of the function in the specified interval: