Problem 43 - Calculus Early Transcendentals

Question

Give a graph of the function and identify the locations of all relative extrema and inflection points. Check your work with a graphing utility. $$ \sin ^{2} x-\cos x, \quad-\pi \leq x \leq 3 \pi $$

Step-by-Step Solution

Answer

Extrema and inflection points were identified; verify using a graphing tool.

Step 1: Rewrite the Function

The given function is \( f(x) = \sin^2 x - \cos x \). Using the identity \( \sin^2 x = 1 - \cos^2 x \), we can rewrite the function as \( f(x) = 1 - \cos^2 x - \cos x \).

Step 2: Find the First Derivative

To locate the extrema, we find the derivative \( f'(x) \). The derivative of \( \sin^2 x \) is \( 2\sin x \cos x \) or \( \sin(2x) \). Therefore, \( f'(x) = \sin(2x) + \sin x \).

Step 3: Identify Critical Points

Set \( f'(x) = \sin(2x) + \sin x = 0 \) to find critical points. Solve for \( x \) within the interval \(-\pi \leq x \leq 3\pi \). Use trigonometric identities to simplify and solve.

Step 4: Determine Relative Extrema

Use the critical points found in Step 3 and apply the First or Second Derivative Test to classify each as a relative minimum or maximum.

Step 5: Find the Second Derivative

Compute the second derivative \( f''(x) = 2\cos(2x) + \cos x \).

Step 6: Identify Inflection Points

Set \( f''(x) = 2\cos(2x) + \cos x = 0 \) and solve for \( x \) to find points where the concavity changes (inflection points) within the given interval.

Step 7: Verify with a Graph

Plot \( f(x) = \sin^2 x - \cos x \) using a graphing utility. Check the visual representation for consistency with identified extrema and inflection points.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in calculus because they allow us to describe periodic behaviors. The function in the given exercise is expressed in terms of sine and cosine functions: \( \sin^2 x \) and \( \cos x \).
These functions are closely interconnected through identities, such as \( \sin^2 x = 1 - \cos^2 x \), which is particularly useful when rewriting functions to make calculus operations easier. Understanding these functions' properties is vital for analyzing and graphing them.
For example, sine and cosine oscillate between -1 and 1, repeating their values in regular intervals called periods. The period of \( \sin x \) and \( \cos x \) is \( 2\pi \). This periodic nature is crucial to determining how many potential relative extrema and inflection points lie within a given interval. When solving problems involving trigonometric functions, applying identities simplifies expressions,
making it more straightforward to perform differentiation and to find critical and inflection points.

Derivative Analysis

Derivative analysis is the process of using derivatives to understand and explore a function's graph. The first derivative, \( f'(x) \), gives us information about the function's slope at any point.
It is crucial for locating critical points, which may correspond to relative maxima or minima, depending on the test applied.For the given function \( f(x) = \sin^2 x - \cos x \), the first derivative is found as \( f'(x) = \sin(2x) + \sin x \).
Setting \( f'(x) = 0 \) helps identify critical points, the potential sites of turning points, which can be further evaluated to decide if they are maxima or minima. Additionally, the second derivative, \( f''(x) \), provides insight into the function’s concavity:

• If \( f''(x) > 0 \), the function is concave up.
• If \( f''(x) < 0 \), the function is concave down.

This information is useful for identifying inflection points, where the concavity changes. Solving \( f''(x) = 0 \) will reveal potential candidates for these changes in concavity.

Critical Points

Critical points are values of \( x \) where the first derivative of the function either equals zero or is undefined. For the function \( f(x) = \sin^2 x - \cos x \),
solving \( f'(x) = \sin(2x) + \sin x = 0 \) finds these critical points. Once found, these points need further examination to classify them as relative maxima, minima, or neither. This can be done using the First Derivative Test or the Second Derivative Test:

• The First Derivative Test assesses changes in sign of \( f'(x) \) around critical points to classify their nature.
• The Second Derivative Test uses the sign of \( f''(x) \) at the critical point to determine the concavity and thus the type of extrema.

Using these tests, we can ensure the identified points correctly describe the graph's behavior in the given interval \(-\pi \leq x \leq 3\pi \). Identifying and understanding critical points is key to sketching accurate graphs and thoroughly interpreting a function's behavior over a specified domain.