Problem 14 - University Calculus: Early Transcendentals

Question

Answer the following questions about the functions whose derivatives are given. a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$f^{\prime}(x)=(\sin x+\cos x)(\sin x-\cos x), 0 \leq x \leq 2 \pi$$

Step-by-Step Solution

Answer

Critical points: \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \). Increasing: \((0, \frac{\pi}{4}), (\frac{3\pi}{4}, \frac{5\pi}{4}), (\frac{7\pi}{4}, 2\pi)\). Local maxima: \( \frac{\pi}{4}, \frac{5\pi}{4} \), minima: \( \frac{3\pi}{4}, \frac{7\pi}{4} \).

Step 1: Simplify the Derivative

We need to simplify the given derivative function \( f'(x) = (\sin x + \cos x)(\sin x - \cos x) \). Use the identity for the difference of squares: \((a+b)(a-b) = a^2 - b^2\).Thus, \[f'(x) = \sin^2 x - \cos^2 x.\]

Step 2: Set Derivative to Zero to Find Critical Points

To find critical points, solve the equation \(f'(x) = 0\).\(\sin^2 x - \cos^2 x = 0\) implies \(\sin^2 x = \cos^2 x\).This can be rewritten as \(\tan^2 x = 1\), leading to \(\tan x = \pm 1\).The solutions within \(0 \leq x \leq 2\pi\) are \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\).

Step 3: Analyze Intervals for Increasing and Decreasing

Evaluate the sign of \(f'(x) = \sin^2 x - \cos^2 x\) on the intervals between critical points.- For \(x \in (0, \frac{\pi}{4})\), test a point like \(x = \frac{\pi}{6}\): \(\sin^2 x - \cos^2 x > 0\), so \(f\) is increasing.- For \(x \in (\frac{\pi}{4}, \frac{3\pi}{4})\), test \(x = \frac{\pi}{2}\): \(\sin^2 x - \cos^2 x < 0\), so \(f\) is decreasing.- For \(x \in (\frac{3\pi}{4}, \frac{5\pi}{4})\), test \(x = \pi\): \(\sin^2 x - \cos^2 x > 0\), so \(f\) is increasing.- For \(x \in (\frac{5\pi}{4}, \frac{7\pi}{4})\), test \(x = \frac{3\pi}{2}\): \(\sin^2 x - \cos^2 x < 0\), so \(f\) is decreasing.- For \(x \in (\frac{7\pi}{4}, 2\pi)\), test \(x = \frac{15\pi}{8}\): \(\sin^2 x - \cos^2 x > 0\), so \(f\) is increasing.

Step 4: Determine Local Maximum and Minimum Points

Based on the sign changes of \(f'(x)\):- At \(x = \frac{\pi}{4}\), \(f(x)\) changes from increasing to decreasing, giving a local maximum.- At \(x = \frac{3\pi}{4}\), \(f(x)\) changes from decreasing to increasing, giving a local minimum.- At \(x = \frac{5\pi}{4}\), \(f(x)\) changes from increasing to decreasing, giving a local maximum.- At \(x = \frac{7\pi}{4}\), \(f(x)\) changes from decreasing to increasing, giving a local minimum.

Key Concepts

Increasing and Decreasing Functions

Understanding where a function is increasing or decreasing is fundamental in analyzing its behavior. A function is said to be **increasing** on an interval if the derivative of the function is positive within that interval. Conversely, a function is **decreasing** when its derivative is negative.

For the given derivative function, we had:

• When evaluating on different intervals between critical points, such as for \(x \in (0, \frac{\pi}{4})\), after testing with a point within this interval, it showed **positive** values, indicating the function is **increasing**.
• In contrast, for \(x \in (\frac{\pi}{4}, \frac{3\pi}{4})\), the evaluation results in **negative** values singifying the function is **decreasing**.

Thus, by analyzing the sign of the derivative, one can categorize intervals of increase and decrease on the entire domain, enabling clear understanding of the function's behavior. Observing these changes helps us determine where the function has peaks or dips, often correlating with our next topic, local maxima and minima.

Local Maximum and Minimum

Local maxima and minima refer to the highest or lowest points in a certain neighborhood around that point. In other words, they are the topmost or bottommost points in those regions that are surrounded by other points with smaller or larger values, respectively.
For any critical point \(x\), if the function changes from increasing to decreasing, there's a **local maximum**. Conversely, if it changes from decreasing to increasing, there's a **local minimum**.

For example:

• At \(x = \frac{\pi}{4}\), since \(f'(x)\) changes from positive to negative, this indicates a local maximum.
• At \(x = \frac{3\pi}{4}\), \(f'(x)\) switches from negative to positive, indicating a local minimum.

Recognizing these points is crucial because local maxima and minima give insight into the structure of a graph and provide valuable information in fields like engineering, economics, and physical sciences where optimization problems are prevalent.

Trigonometric Derivatives

Trigonometric functions, such as sine and cosine, have specific derivatives that we often utilize in calculus. Knowing these derivatives helps us discover other characteristics of a function, as seen in the provided function:
\[f'(x) = \sin^2 x - \cos^2 x.\]This result is derived from the trigonometric identity for the difference of squares \((a^2 - b^2)\), where :

• The derivative of \(\sin x\) is \(\cos x\)
• The derivative of \(\cos x\) is \(-\sin x\)

These derivatives play a vital role in finding critical points, understanding changes in intervals, and comprehending local extrema.

Solving problems related to trigonometric derivatives requires practice and a good grasp of trigonometric identities. Such knowledge adds efficiency in determining function properties and contributes to an understanding of rotational or oscillatory motions, often modeled by trigonometric functions.