Problem 82 - Precalculus with Limits

Question

In Exercises \(79-84,\) (a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f\) . (Calculus is required to find the trigonometric equation.) $$\begin{array}{ll} \qquad {\text { Function }} & {\text { Trigonometric Equation }} \\ {f(x)=2 \sin x+\cos 2 x} & {2 \cos x-4 \sin x \cos x=0}\end{array}$$

Step-by-Step Solution

Answer

The maximum and minimum points of the function \(f(x) = 2 \sin x + \cos 2x \) in the interval [0,2) are approximately given by the solutions to the equation \( 2 \cos x - 4 \sin x \cos x = 0 \). Exact values will depend on the specific solutions found.

Step 1: Graph the function

Use a graphing tool to plot the function \(f(x) = 2 \sin x + \cos 2x \) on the interval [0,2). Look for peak and trough points which represent maximum and minimum points.

Step 2: Approximate max/min points

Upon graphing, approximate the x-values at which the maximum and minimum occurs within the interval [0,2). Let's denote these values as \(x_1\) for the min point, and \(x_2\) for the max point. Remember these are just approximations now as we will solve the trigonometric equation to find the accurate values.

Step 3: Solve the trigonometric equation

Solve the given equation \( 2 \cos x - 4 \sin x \cos x = 0 \). This equation can be factorized into \(2 \cos x (1 - 2 \sin x) = 0\), giving \(\cos x = 0\) and \(\sin x = 1/2\) as the solutions that correspond to the x-coordinates for the max and min points of \(f\). Note that we are interested in solutions in the interval [0,2).

Step 4: Compare values

Compare the exact solutions of the trigonometric equation with the approximated values obtained from the graph. They should correspond to each other, confirming that the solutions to the equation indeed are the x-coordinates of the max and min points of the function \(f\).