Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=\frac{x^{3}}{3}+2 x+\frac{3}{x}\) on \((0,3]\)

Step-by-Step Solution

Answer

To solve this, calculate the first derivative and find the values for which it is zero or undefined. This will give the critical points. Then calculate the second derivative and substitute the critical points into this to classify them. Finally, compare the function values at the critical points and at the ends of the interval to find the absolute maximum and minimum.

Step 1: Calculate the Derivative of the Function

The derivative of \(f(x)\) is given as \(f'(x)\). This is calculated using the power rule for derivatives. \n So, \(f'(x)=x^{2}+2-\frac{3}{x^{2}}\).

Step 2: Find Critical Points

critical points are found by setting \(f'(x)\) to zero and by identifying where \(f'(x)\) is undefined. In this case, \(f'(x)\) is undefined for \(x=0\) which is not a point in our interval, so we ignore this. Solving \(f'(x)=0\), we get \(x^{2}+2-\frac{3}{x^{2}}=0\). The solutions to this will provide the critical points.

Step 3: Classify Each Critical Point

When we find the second derivative, and substitute the critical points from step 2, we can classify the critical points. If the second derivative \(f''(x)\) is positive, the critical point is a local minimum, if \(f''(x)\) is negative, the critical point is a local maximum, if \(f''(x)\) doesn't exist or equal zero, critical point is an inflection point. To classify as an absolute max or min, it will depend on the function values at the critical points and at the interval's endpoints.

Step 4: Find Absolute Max and Min Values

We can find the absolute maximum and minimum values by comparing the function values at the critical points and at the ends of the specified interval. The highest of these values will be the maximum and the lowest will be the minimum.

Key Concepts

Derivative of a Function

The derivative of a function is a fundamental concept in calculus. It measures how a function changes as its input changes. The derivative is denoted as \( f'(x) \) for a function \( f(x) \). In our example, we start by computing the derivative of \( f(x)=\frac{x^{3}}{3}+2 x+\frac{3}{x} \) on the interval \((0,3]\). By applying the power and quotient rule, the derivative is given by \( f'(x) = x^{2} + 2 - \frac{3}{x^{2}} \). This derivative helps us understand where the function is increasing or decreasing.

• When \( f'(x) > 0 \), the function is increasing.
• When \( f'(x) < 0 \), the function is decreasing.
• Critical points occur where \( f'(x) = 0 \) or \( f'(x) \) is undefined.

These critical points are significant in determining the behavior of the function across the specified interval.

Local Maximum and Minimum

Local maxima and minima are points on the graph of a function where that function reaches local highs or lows. These are detected by examining critical points where \( f'(x) = 0 \). A point \( x = c \) is a local maximum if \( f(x) \geq f(x) \) for all \( x \) in a neighborhood of \( c \), and a local minimum if \( f(x) \leq f(x) \) nearby.
The second derivative \( f''(x) \) helps classify whether we experience a max or min at these points:

• If \( f''(x) > 0 \), the point is a local minimum.
• If \( f''(x) < 0 \), the point is a local maximum.
• If \( f''(x) = 0 \) or does not exist, the test is inconclusive.

Applying these tests helps us confirm whether a critical point is a local minimum or maximum.

Absolute Maximum and Minimum

Finding the absolute maximum and minimum of a function involves identifying the largest and smallest function values in the entire interval. Absolute extrema are not limited to a neighborhood but across the entire domain of interest. In practical terms, this means evaluating the function at:

• All critical points within the interval.
• The boundaries of the interval (in our example, at \( x = 3 \)).

Compare these values to determine:

• The absolute maximum, which is the highest value.
• The absolute minimum, which is the lowest value.

This process ensures we find the points where the function reaches its highest and lowest values over the specified domain.

Second Derivative Test

The second derivative test is a method used to find and classify the local maxima and minima of a function based on its second derivative. Once you have critical points from \( f'(x) = 0 \), apply the second derivative test as follows:

• Calculate the second derivative \( f''(x) \).
• Substitute each critical point into \( f''(x) \).

The outcomes tell us about the concavity of the function:

• If \( f''(x) > 0 \) at a critical point, the function is concave up, indicating a local minimum.
• If \( f''(x) < 0 \), the function is concave down, meaning a local maximum.
• If \( f''(x) = 0 \), reevaluate with another method because the test is inconclusive.

Understanding this process helps to accurately classify the nature of critical points we previously identified.