Problem 32

Question

Find a good bound for the maximum value of the given expression, given that $c$ is in the stated interval. Answers may vary depending on the technique used. (See Example 5.) $$ \left|\frac{4 c}{c+4}\right| ;[0,1] $$

Step-by-Step Solution

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Answer

The maximum bound for $ \left|\frac{4c}{c+4}\right| $ over $[0,1]$ is $ \frac{4}{5} $.

1Step 1: Analyze the Expression

We're given the expression $ \left|\frac{4c}{c+4}\right| $ and asked to find a bound for its maximum value for $ c $ in the interval $[0,1]$. The expression involves a rational function within absolute value.

2Step 2: Plug in Endpoints of the Interval

Evaluate the expression at the endpoints of the interval, $c = 0$ and $c = 1$. Calculating:- When $c=0$, $\left|\frac{4 \times 0}{0 + 4}\right| = 0$.- When $c=1$, $\left|\frac{4 \times 1}{1 + 4}\right| = \left|\frac{4}{5}\right| = \frac{4}{5}$.

3Step 3: Consider the Behavior Between the Endpoints

The function $ \frac{4c}{c+4} $ is continuous on $[0,1]$. Calculate its derivative to analyze where it may attain a maximum. The derivative is given by: \[f'(c) = \frac{d}{dc}\left(\frac{4c}{c+4}\right) = \frac{4(c+4) - 4c}{(c+4)^2} = \frac{16}{(c+4)^2}.\] Since $ f'(c) > 0 $ for all $ c $ in $[0, 1]$, the function is increasing on the interval.

4Step 4: Determine the Maximum Value in the Interval

Since the function is increasing and continuous over $[0,1]$, its maximum value will occur at the highest value of $c$, which is $c = 1$. The expression value at this point is $ \frac{4}{5} $ as calculated in Step 2.

Key Concepts

Rational FunctionsMaximum ValueIntervalsContinuous Functions

Rational Functions

Rational functions are a key concept in calculus and consist of fractions that have polynomials in both the numerator and the denominator. In our given exercise, the expression $ \frac{4c}{c+4} $ is such a rational function. Understanding these expressions is crucial because they allow us to model real-world problems with variables that can change. Rational functions can have important characteristics like asymptotes, which are lines that the graph of the function approaches but never touches. While our specific function does not have vertical asymptotes for $ c $ in the interval $[0, 1]$, it is worth noting that analyzing these potential features can affect how we find the maximum values of such functions.

Maximum Value

The maximum value of a function on a given interval is the greatest y-value it can achieve there. It requires evaluating endpoints and any critical points where the derivative equals zero or does not exist, which can indicate local maxima. In our rational function $ \left|\frac{4c}{c+4}\right| $, to find the maximum value, we first checked the endpoints of the interval— for $ c=0 $ and $ c=1 $.

When $ c=0 $, the expression equals 0.
When $ c=1 $, it equals $ \frac{4}{5} $.

Then, by calculating the derivative $ f'(c) $, it was observed that $ f'(c) > 0 $ over the entire interval. This shows the function is continuously increasing, meaning the maximum value in the interval is reached at $ c=1 $. Thus, $ \frac{4}{5} $ is indeed the maximum here.

Intervals

Intervals define the ranges of values for which we are interested in the behavior of a function. For our problem, the interval $[0, 1]$ specifies the range of $ c $ values that we are considering in order to find the maximum value.Understanding intervals is crucial because they help us to limit our focus only to relevant parts of the function where certain conditions might hold, like finding where it may reach a peak or valley in a specified range.Working within established interval limits avoids trying to analyze the entire domain of a complex function, allowing clearer focus and faster problem-solving.

Continuous Functions

Continuous functions are those that have no breaks, jumps, or holes in their graph within a certain range. Calculus heavily relies on functions being continuous, as it allows for the application of derivatives and integrals. In this exercise, $ \frac{4c}{c+4} $ is continuous across the interval $[0,1]$. This continuity helped ensure that the increasing nature of the derivative $ f'(c) $ automatically led to determining the maximum at the endpoint.Continuous functions are integral in ensuring smooth operations and analysis like predicting behavior, anticipating maximum or minimum values, and confirming certain properties or formulas within a given range.

Problem 31

Problem 32

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