Problem 126 - Thomas Calculus

Question

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\)-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? \begin{equation}\begin{array}{l}{\text { Graph } f(x)=x \cos x \text { and its second derivative together for }} \\ {0 \leq x \leq 2 \pi . \text { Comment on the behavior of the graph of } f \text { in }} \\ {\text { relation to the signs and values of } f^{\prime \prime} \text {.}}\end{array}\end{equation}

Step-by-Step Solution

Answer

The critical points and inflection points found by solving \(f'(x)=0\) and \(f''(x)=0\) determine where the function changes extremum or concavity. Graphs at these points verify this behavior.

Step 1: Find the First Derivative

Calculate the first derivative of the function \( f(x) = x \cos x \) using the product rule. Let \( u = x \) and \( v = \cos x \), then \( u' = 1 \) and \( v' = -\sin x \). Using the product rule, \( f'(x) = u'v + uv' = -x \sin x + \cos x \).

Step 2: Find Critical Points

To find critical points, solve \( f'(x) = -x \sin x + \cos x = 0 \). This simplifies to \( \cos x = x \sin x \). Numerical or graphical methods will help solve for \( x \), as there is no simple algebraic solution.

Step 3: Find the Second Derivative

Now, find the second derivative of \( f(x) \, namely, \, f''(x) \). Differentiate \( f'(x) = -x \sin x + \cos x \) using the product rule: \( f''(x) = -(x \cos x + \sin x) - \sin x = -x \cos x - 2\sin x \).

Step 4: Determine Inflection Points

Inflection points occur where \( f''(x) = 0 \). Solve \( f''(x) = -x \cos x - 2\sin x = 0 \). Set \( g(x)= -x \cos x - 2 \sin x \) and use numerical or graphical methods to solve \( g(x) = 0 \).

Step 5: Analyze Functions and Derivatives

Graph \( f(x), \ f'(x) \text{ and } f''(x) \) over the interval \([0, 2\pi]\). Observe where \( f'(x) \) and \( f''(x) \) intersect the x-axis, indicating critical and inflection points respectively.

Step 6: Verify and Describe Relationships

From the graph, identify where \( f'(x) = 0 \) correlates to peaks and troughs in \( f(x) \) and where \( f''(x) = 0 \) indicates changes in concavity. Determine if \( f \) has a local max or min by checking sign changes in \( f'(x) \).

Step 7: Behavior of the Graph

Based on the graphs, local maxima or minima occur where \( f'(x) = 0 \). Inflection points occur where \( f''(x) = 0 \) and the concavity changes. Relate intervals of increasing or decreasing \( f(x) \) with the sign of \( f'(x) \), and concavity with \( f''(x) \).

Key Concepts

Critical Points

Critical points in calculus are where the first derivative of a function is zero or undefined. They are important as they often indicate potential local maxima, minima, or saddle points. To find these points for a function like \(f(x) = x\cos x\), we look at its first derivative \(f'(x) = -x \sin x + \cos x\) and solve \(f'(x) = 0\). This means finding values of \(x\) where the slope of the tangent is horizontal (i.e., the function's rate of change is temporarily paused).
These critical points do not guarantee a peak or trough but are candidates that need further verification, often by using the second derivative or examining the sign of the first derivative around these points. Thus, solving \(-x \sin x + \cos x = 0\) is a crucial step toward locating critical points.

Second Derivative

The second derivative of a function gives us information about the curvature of the function—how it bends at different points. For our specific function, the second derivative is \(f''(x) = -x \cos x - 2 \sin x\).
The places where \(f''(x) = 0\) are potential inflection points, where the concavity of the function might change. The second derivative can also help classify the critical points found from the first derivative.

• If \(f''(x) > 0\), the function is concave up, and if there was a critical point at that \(x\), it might be a local minimum.
• If \(f''(x) < 0\), the function is concave down, and a critical point could be a local maximum.

Examining the sign of \(f''(x)\) gives deep insight into the function's overall shape and dynamics.

Concavity

Concavity relates to whether a function is bending upwards or downwards. It provides invaluable context about the function's growth direction between known points.

• A function is concave up if its graph looks like a cup (i.e., \(f''(x) > 0\)). This suggests intervals where the function is accelerating upwards, potentially leading to local minima.
• Conversely, a function is concave down if its graph opens like a cap (i.e., \(f''(x) < 0\)). Such intervals often indicate slowing down, pointing towards possible local maxima.

By examining \(f'(x)\) in these intervals, we can better predict whether the function is increasing or decreasing. This duality between concavity and the first derivative informs us about the full curvature dynamics, making it clearer to spot local extrema or inflection points.

Local Maximum

A local maximum is a point where the function reaches a peak relative to points nearby. Mathematically, it can be verified by using the first and second derivatives.
In simple words:

• If \(f'(x) = 0\) at \(x = c\) and \(f''(c) < 0\), then \(f\) has a local maximum at \(x = c\).
• This occurs because the function changes from increasing to decreasing, with the slope reversing direction sharply.

Effectively identifying local maxima allows us to pinpoint where a function briefly hits its highest value and then turns downward. Understanding these points can be invaluable, especially in real-world applications where peaks represent optimal values, such as maximizing profit or efficiency.

Local Minimum

Local minima are the lowest points on a function within a particular region. They are not necessarily the absolute lowest point of the function but signify a bottom or valley compared to immediate surroundings.
To identify a local minimum:

• Check where the first derivative \(f'(x) = 0\) or changes sign.
• If \(f''(x) > 0\) at that critical point, the point is likely a local minimum, because it indicates the function changes from decreasing to increasing, a shift that embodies a trough.

In practical terms, local minima are crucial for recognizing minimum cost, risk, or time in various applications, representing the most desirable point within a restricted frame.