I am having some trouble with the following question:
Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum (or neither). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
Function: f(x) = 4 sin x cos x, on (0, π)
I was successfully able to get the local minimum and local maximum for this function which are:
3π/4 (local min)
π/4 (local max)
However, I have no idea what to do for the following:
Determine the intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. Enter EMPTY or ∅ for the empty set.)
Any help is greatly appreciated.
Best Answer
Hint
You have
$$f'(x)=4(\cos^2 x-\sin^2 x).$$ To get the critical points you have to solve $f'(x)=0.$ You have done it and you have obtained $x=\pi/4$ and $x=3\pi/4.$ Now, you have to study the sign of $f$ on the intervals $(0,\pi/4),$ $(\pi/4,3\pi/4)$ and $(3\pi/4,\pi).$ ($0$ and $\pi$ because they are the extremes of the interval where the function is defined and $x\pi/4$ and $3\pi/4$ because they are the critical points.)
Remember that if $f'(x)>0,\: x\in (a,b)$ then $f$ is strictly increasing in $(a,b)$ and if $f'(x)<0,\: x\in (a,b)$ then $f$ is strictly decreasing in $(a,b).$ A local minimum is obtained when you change from an interval where the function is decreasing to an interval where it is increasing. It is similar for a local maximum. (This is the first derivative test.)
Can you finish?