Problem 48 - Essential Calculus

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. $$ f(x)=\sin x \cos x $$

Step-by-Step Solution

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 1: 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 \(\cos^2 x - \sin^2 x\).

Step 2: 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 \(x = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4\).

Step 3: 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 \((5\pi/4, 7\pi/4)\).

Step 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 \(x = 3\pi/4, 7\pi/4\).

Step 5: 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.

Key Concepts

Critical Points

Critical points are very important in calculus because they help us understand the behavior of functions. A critical point occurs where the derivative of a function is zero or undefined. For the function \( f(x) = \sin x \cos x \), the derivative \( f'(x) \) is found to be \( \cos^2 x - \sin^2 x \). To find the critical points, we set this derivative equal to zero and solve for \( x \). By doing so, we get four critical points: \( x = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4 \). These points are special because they indicate where the function might change from increasing to decreasing, or vice versa.
Finding critical points is crucial because they are often where relative maxima or minima occur. They help us to later determine intervals where the function increases or decreases and to locate points where the slope is zero.

Increasing and Decreasing Intervals

Increasing and decreasing intervals show us where a function goes up or down. To determine if a function is increasing or decreasing on an interval, we use its derivative. When the derivative is positive over an interval, the function is increasing there. If it's negative, the function is decreasing.
For \( f(x) = \sin x \cos x \), and using its derivative \( f'(x) = \cos^2 x - \sin^2 x \), we test intervals around the critical points \( \pi/4, 3\pi/4, 5\pi/4, 7\pi/4 \).

• The function is increasing between \( (0, \pi/4) \) and \( (3\pi/4, 5\pi/4) \).
• The function is decreasing between \( (\pi/4, 3\pi/4) \) and \( (5\pi/4, 7\pi/4) \).

These intervals are important because they show us where the function is climbing or descending, helping us understand its overall shape.

Relative Extrema

Relative extrema are points where a function reaches a local maximum or minimum within a given interval. To find relative extrema, the First Derivative Test is very useful. It involves analyzing the sign changes of the derivative around critical points.
When applying the First Derivative Test to \( f(x) = \sin x \cos x \), we note:

• A positive to negative change at \( x = \pi/4 \) and \( 5\pi/4 \) indicates a local maximum.
• A negative to positive change at \( x = 3\pi/4 \) and \( 7\pi/4 \) indicates a local minimum.

This test helps us pinpoint exactly where the function obtains these high and low points, giving us a clearer picture of the function's graph and behavior. Such points are useful in understanding the function's most significant points without analyzing the entire graph.