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. $$ f(x)=\cos ^{2}(2 x) $$

Short answer

The function \(f(x)=\cos^{2}(2x)\) is increasing on the intervals \((\frac{\pi}{4}, \frac{3\pi}{4})\) and \((\frac{5\pi}{4}, \frac{7\pi}{4})\), and decreasing on intervals \((0, \frac{\pi}{4})\), \((\frac{3\pi}{4}, \frac{5\pi}{4})\) and \((\frac{7\pi}{4}, 2\pi)\). The function has relative maxima at \(x=\frac{\pi}{4}\), \(x=\frac{5\pi}{4}\), and relative minima at \(x=\frac{3\pi}{4}\), \(x=\frac{7\pi}{4}\).

Step-by-step solution

Find the derivative of the function

The derivative of a function gives us the slope of the function at any point. We first need to find the derivative of the function \(f(x)=\cos^{2}(2x)\). Using the chain rule for differentiation, the derivative of \(f(x)=\cos^{2}(2x)\) is \(f'(x)=-4\cos(2x)\sin(2x)\).

Find the critical points

Critical points occur where the derivative is zero or undefined. We set \(f'(x)=0\), resulting in \(-4\cos(2x)\sin(2x)=0\). Solving this we get \(x=\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) within the interval \((0,2 \pi)\), wherein the function changes from increasing to decreasing or vice versa. These are critical points.

Determine intervals of increase and decrease

We choose test points in each interval determined by the critical points and evaluate the sign of the derivative at these test points. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing. Test points could be \(\frac{\pi}{8}, \frac{\pi}{2}, \frac{9\pi}{8}, \frac{5\pi}{2}\). We observe that for these points, \(f'(x)>0\) when \(x \in (\frac{\pi}{4}, \frac{3\pi}{4})\) and \(x \in (\frac{5\pi}{4}, \frac{7\pi}{4})\) (function is increasing), and \(f'(x)<0\) when \(x \in (0, \frac{\pi}{4})\) and \(x \in (\frac{3\pi}{4}, \frac{5\pi}{4})\) and \(x \in (\frac{7\pi}{4}, 2\pi)\) (function is decreasing).

Apply the First Derivative Test

The first derivative test is used to determine where the function has its relative extrema. When the function changes from increasing to decreasing at a critical point, that point is a relative maximum. When the function changes from decreasing to increasing at a critical point, that point is a relative minimum. Using this, we find that \(x=\frac{\pi}{4}\) and \(x=\frac{5\pi}{4}\) are relative maxima, and \(x=\frac{3\pi}{4}\) and \(x=\frac{7\pi}{4}\) are relative minima.

Confirm with a Graph

Lastly, it's recommended to use a graphing tool to visually confirm the intervals of increase and decrease, and the locations of relative extrema. When the function is graphed, it will show increases and decreases consistent with our findings, and relative extrema at the identified points.