Question

Consider the function on the interval \((0,2 \pi)\) For each function, (a) find the open interval(s) on which the function is increasing or decreasing, (b) apply the First Derivative Test to identify all relative extrema, and (c) use a graphing utility to confirm your results. $$ f(x)=\sin x \cos x $$

Short answer

The function \(f(x) = \sin x \cos x\) is increasing on the intervals \((0, \pi/4)\) and \((3\pi/4, 5\pi/4)\), and decreasing on the intervals \((\pi/4, 3\pi/4)\) and \((5\pi/4, 7\pi/4)\). The function has maximum points at \(x = \pi/4, 5\pi/4\) and minimum points at \(x = 3\pi/4, 7\pi/4\).

Step-by-step solution

Calculate the Derivative

Start by computing the derivative of the function \(f(x) = \sin x \cos x\). We will use the product rule which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. For \(f(x) = \sin x \cos x\), the derivative \(f'(x)\) is \(\cos^2 x - \sin^2 x\).

Find the Critical Points

The critical points of a function are found by setting the derivative equal to zero. Solving \(f'(x) = \cos^2 x - \sin^2 x = 0\), we get \(x = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4\).

Determine Increasing and Decreasing Intervals

We test the intervals between the critical points to determine whether the function is increasing or decreasing in that interval. From the interval test, we know that \(f(x)\) is increasing on the intervals \((0, \pi/4)\) and \((3\pi/4, 5\pi/4)\), and decreasing on the intervals \((\pi/4, 3\pi/4)\) and \((5\pi/4, 7\pi/4)\).

Apply the First Derivative Test

We use the First Derivative Test to find out the relative extrema. A positive to negative change indicates a local maximum, whereas a negative to positive change indicates a local minimum. Therefore, we have maximum points at \(x = \pi/4, 5\pi/4\) and minimum points at \(x = 3\pi/4, 7\pi/4\).

Graph the Function

Confirm these findings by graphing the function \(f(x)\). The graph of \(f(x) = \sin x \cos x\) on the interval \((0, 2\pi)\) confirms that the trends and extrema found coincide with the graph.