Problem 55 - Thomas Calculus

Question

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}. \) $$ f(x)=\sqrt{3} \cos x+\sin x, \quad 0 \leq x \leq 2 \pi $$

Step-by-Step Solution

Answer

Local maximum at \(x = \frac{\pi}{6}\) and local minimum at \(x = \frac{7\pi}{6}\). \( f \) increases where \( f' > 0 \) and decreases where \( f' < 0 \).

Step 1: Find the Derivative

To find the local extrema, first calculate the derivative of the function \( f(x) = \sqrt{3} \cos x + \sin x \). The derivative is \( f'(x) = -\sqrt{3} \sin x + \cos x \).

Step 2: Determine Critical Points

Set the derivative equal to zero to find the critical points. Solve \( -\sqrt{3} \sin x + \cos x = 0 \). This simplifies to \( \cos x = \sqrt{3} \sin x \), so \( \tan x = \frac{1}{\sqrt{3}} \). The angles that satisfy this equation within the interval \([0, 2\pi]\) are \(x = \frac{\pi}{6}\) and \(x = \frac{7\pi}{6}\).

Step 3: Evaluate the Function at Critical Points and Endpoints

Evaluate \( f(x) \) at \(x = 0\), \(x = \frac{\pi}{6}\), \(x = \frac{7\pi}{6}\), and \(x = 2\pi\). Calculate: \( f(0) = \sqrt{3}, \ f\left(\frac{\pi}{6}\right) = \sqrt{3}\cos\left(\frac{\pi}{6}\right) + \sin\left(\frac{\pi}{6}\right) = 2, \ f\left(\frac{7\pi}{6}\right) = \sqrt{3}\cos\left(\frac{7\pi}{6}\right) + \sin\left(\frac{7\pi}{6}\right) = -2, \ f(2\pi) = \sqrt{3} \).

Step 4: Identify Local Extrema

The highest value is \( f\left(\frac{\pi}{6}\right) \) which is 2 (local maximum), and the lowest value is \( f\left(\frac{7\pi}{6}\right) \) which is -2 (local minimum) within the interval \([0, 2\pi]\).

Step 5: Graph Function and Derivative

Graph \( f(x) = \sqrt{3} \cos x + \sin x \) and its derivative \( f'(x) = -\sqrt{3} \sin x + \cos x \). Observe that \( f'(x) > 0 \) indicates \( f(x) \) is increasing, and \( f'(x) < 0 \) indicates \( f(x) \) is decreasing. The sign changes of \( f'(x) \) at critical points confirm extreme values.

Step 6: Analyze Behavior and Comment

Where \( f'(x) > 0 \) (between \(x = 0\) and \(x = \frac{\pi}{6}\), \(x = \frac{7\pi}{6}\) and \(x = 2\pi\)),\( f(x) \) increases, while \( f(x) \) decreases where \( f'(x) < 0 \) (between \(x = \frac{\pi}{6}\) and \(x = \frac{7\pi}{6}\)). This verifies our critical points and shows how \( f(x) \) behaves in relation to \( f'(x) \).

Key Concepts

Derivative

To find local extrema, the first crucial step involves determining the derivative of the function. The derivative provides vital information about the rate of change or slope of the function at any given point. In mathematical notation, it is usually denoted by \( f'(x) \). For example, when differentiating the function \( f(x) = \sqrt{3} \cos x + \sin x \), we obtain the derivative \( f'(x) = -\sqrt{3} \sin x + \cos x \).
This derivative helps identify whether the original function is increasing or decreasing at specific points.
An important point to note: whenever \( f'(x) \) is positive, the function is increasing, and whenever \( f'(x) \) is negative, it's decreasing.

• Calculating derivatives involves understanding how to apply rules and formulas such as the chain rule, power rule, and product rule.
• In trigonometric functions, recognizing standard derivatives of \( \sin x \), \( \cos x \), and \( \tan x \) is essential.

Mastering derivatives helps in predicting the function's behavior without necessarily plotting the entire graph.

Critical Points

Critical points play a pivotal role in identifying local extrema, such as maximums and minimums, within a given interval. These occur where the derivative equals zero or is undefined. For the function \( f(x) = \sqrt{3} \cos x + \sin x \), we set the derivative \( f'(x) = -\sqrt{3} \sin x + \cos x \) equal to zero to find critical points. This simplifies to solving \( \tan x = \frac{1}{\sqrt{3}} \).
The solutions within the interval \([0, 2\pi]\) give the values \( x = \frac{\pi}{6} \) and \( x = \frac{7\pi}{6} \).
By testing the endpoints and these critical values within the interval, students can find where extrema occur.

• Critical points are not always points of extrema; they merely suggest locations where extrema might occur.
• Understanding how to solve trigonometric equations is key to finding the right critical points.

Using the derivative to find critical points is fundamental in understanding the deeper structure of the function.

Trigonometric Functions

Trigonometric functions, such as \( \sin x \), \( \cos x \), and \( \tan x \), are periodic and play a fundamental role in analyzing functions concerning angles and cycles. In this context, the function \( f(x) = \sqrt{3} \cos x + \sin x \) uses properties of cosines and sines. Understanding these functions helps in evaluating how they oscillate and intersect within given intervals.
Trigonometric identities, like the tangent equation \( \tan x = \frac{1}{\sqrt{3}} \), simplify solving for critical points and influence the general shape of the graph.

• Each trigonometric function has a specific period and range, influencing how they are plotted.
• Identities and angle transformations ease the complex algebraic manipulations during problem-solving.

Grasping the basics of trigonometric functions aids in predicting the behavior of more complex functions built from them.

Graph Analysis

Graph analysis is essential for visualizing how functions behave over certain intervals. Plotting \( f(x) = \sqrt{3} \cos x + \sin x \) alongside its derivative \( f'(x) = -\sqrt{3} \sin x + \cos x \) helps confirm the analytical work done in finding extrema and critical points.
Observations from the graph confirm where \( f'(x) \) crosses zero, which coincide with critical points and provide a clear depiction of where \( f(x) \) increases or decreases.

• A graph's peak and trough are indicative of local maxima and minima, respectively.
• The slope of the tangent (from the derivative) at various points helps visually confirm increasing and decreasing trends.

Effective graph analysis links visual insight with mathematical formality, enhancing students' understanding of the relationships within functions.