Problem 57 - Essential Calculus Early Transcendentals

Question

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. $$f(x)=6 \sin x+\cot x, \quad-\pi \leqslant x \leqslant \pi$$

Step-by-Step Solution

Answer

Use derivative graphs to find intervals of change, extrema, concavity, and inflection points.

Step 1: Derivative Calculation

To find the intervals of increase and decrease, we need to compute the first derivative of the function. The given function is \( f(x) = 6 \sin x + \cot x \). The derivative \( f'(x) \) is found using the derivatives \( \frac{d}{dx}[\sin x] = \cos x \) and \( \frac{d}{dx}[\cot x] = -\csc^2 x \). So, the derivative is: \[ f'(x) = 6 \cos x - \csc^2 x \].

Step 2: Analyze Critical Points

To find the critical points where the function changes increasing/decreasing behavior, solve \( f'(x) = 0 \): \[ 6 \cos x - \csc^2 x = 0 \].This equation can be solved algebraically or graphically to identify the points. Additionally, check where \( f'(x) \) does not exist, i.e., where \( \csc x \) is undefined.

Step 3: Determine Intervals of Increase/Decrease

Using the critical points found, analyze the sign of \( f'(x) \) on intervals between these points. If \( f'(x) > 0 \), the function is increasing, and if \( f'(x) < 0 \), it is decreasing. Consider the domain \( -\pi \leq x \leq \pi \).

Step 4: Second Derivative Calculation

To find intervals of concavity and inflection points, compute the second derivative \( f''(x) \). The derivative of \( f'(x) = 6 \cos x - \csc^2 x \) is:\[ f''(x) = -6 \sin x + 2 \cot x \csc^2 x \].

Step 5: Identify Concavity Intervals and Inflection Points

Determine where \( f''(x) = 0 \) for potential inflection points. Evaluate the sign of \( f''(x) \) in the intervals determined by these points. If \( f''(x) > 0 \), the function is concave up; if \( f''(x) < 0 \), concave down.

Step 6: Graph Interpretation

Create plots of \( f(x) \), \( f'(x) \), and \( f''(x) \) across the interval \( -\pi \leq x \leq \pi \) to visually interpret the behavior. Identify minimum, maximum, and inflection points, and verify findings from the calculations.

Key Concepts

Derivative Calculations

To understand curve behavior, we begin by computing the derivative of the function. Here, the function is given as \( f(x) = 6 \sin x + \cot x \). The derivative, denoted as \( f'(x) \), represents the slope of the tangent to the curve at any point. Calculating this derivative involves using the known derivatives of \( \sin x \) and \( \cot x \).

The derivative of \( \sin x \) is \( \cos x \), and the derivative of \( \cot x \) is \( -\csc^2 x \). Combining these results, we find:

• Derivative: \( f'(x) = 6 \cos x - \csc^2 x \)

This derivative gives us important insight into the function's behavior, like where it rises or falls.

Critical Points

Critical points of a function occur where the derivative is zero or undefined. These points help us identify where the function changes from increasing to decreasing, or vice versa. To find these points for our function, we set \( f'(x) = 0 \).

Solving \( 6 \cos x - \csc^2 x = 0 \), we identify potent critical points. We should also consider where the function's derivative doesn't exist, specifically where \( \csc x \) (the cosecant function) is undefined.

At critical points, non-differentiability due to values like multiples of \( \pi \) leads to further analysis for complete understanding.

Intervals of Increase and Decrease

Once critical points are identified, they split the domain into intervals. Within each interval, the function is either increasing or decreasing. To determine this, analyze the sign of the derivative, \( f'(x) \).

If \( f'(x) > 0 \) for any interval, the function increases there. Conversely, if \( f'(x) < 0 \), the function decreases. Thus, the sign of \( f'(x) \) on each sub-interval tells us the behavior. Since our domain is from \(-\pi\) to \(\pi\), it's crucial to evaluate behavior close to these endpoints while considering periodic behavior.

This analysis offers clear insights into the function's growth and contraction phases over its defined interval.

Concavity and Inflection Points

Concavity describes how the curve bends over stretches of its graph. It is defined via the second derivative, \( f''(x) \). To establish intervals of concavity, we computed \( f''(x) = -6 \sin x + 2 \cot x \csc^2 x \).

If \( f''(x) > 0 \), the function is concave up (like a cup); if \( f''(x) < 0 \), it's concave down (like a frown). Check where \( f''(x) = 0 \) for potential inflection points, where concavity changes.
Inflection points divide the curve into sections with different concavity states, adding depth to our understanding of its nature. Thus, assessing \( f''(x) \) confirms where the curve undergoes shifts, enriching the graphical interpretation of the function.