Problem 45 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=x-\log _{4} x $$

Step-by-Step Solution

Answer

The critical number of the function \(f(x)\) is \(x=1/\ln(4)\). The function is increasing on the interval \((-\infty, 1/\ln(4))\) and decreasing on the interval \((1/\ln(4), \infty)\). The relative maximum is at \(x=1/\ln(4)\).

Step 1: Calculate derivative

The differentiation of \(x\) is \(1\) and the differentiation of \(-\log_{4}x\) is \(-1/x\ln(4)\). So the derivative of \(f(x)\) is \(f'(x)=1-\frac{1}{x \ln(4)}\).

Step 2: Find critical numbers

To find the critical numbers, set the derivative equal to zero and solve for \(x\). \n\(1-\frac{1}{x \ln(4)}=0\) gives \(x=1/\ln(4)\). The critical number is valid since it lies in the domain of the original function \(f(x)\). So the critical number is \(x=1/\ln(4)\).

Step 3: Test critical number on derivative

We test the critical number in the derivative to find intervals where the function is increasing or decreasing. Choose \(x=0\) for \(x<1/\ln(4)\) and \(x=1\) for \(x>1/\ln(4)\). Substituting these test points in \(f'(x)\) we have, \(f'(0)>0\) and \(f'(1)<0\). So, \(f(x)\) is increasing on \((-\infty, 1/\ln(4))\) and decreasing on \((1/\ln(4), \infty)\).

Step 4: Find relative extrema

As \(f(x)\) changes from increasing to decreasing at \(x=1/\ln(4)\), a relative maxima exists at this point. Hence, the relative maximum is at \(x=1/\ln(4)\).

Key Concepts

Increasing and Decreasing Intervals

Understanding when a function is increasing or decreasing is crucial for analyzing its behavior. To determine this, we inspect the derivative of the function, denoted as \( f'(x) \). The basic idea is straightforward:

• If \( f'(x) > 0 \), then the function is increasing on that interval.
• If \( f'(x) < 0 \), then the function is decreasing on that interval.

We applied this method to our function \( f(x) = x - \log_{4} x \).
By examining the derivative \( f'(x) = 1 - \frac{1}{x \ln(4)} \), we placed test points on either side of the critical number \( x = \frac{1}{\ln(4)} \).
For \( x < \frac{1}{\ln(4)} \), such as \( x = 0.5 \), we found \( f'(x) > 0 \), indicating that our function increases on \((-fty, \frac{1}{\ln(4)})\). Conversely, for \( x > \frac{1}{\ln(4)} \), like \( x = 2 \), we found \( f'(x) < 0 \), indicating a decrease on \((\frac{1}{\ln(4)}, \infty)\). This complete analysis lets us clearly see on which intervals the function rises and falls.

Relative Extrema

Relative extrema refer to points in the domain of a function where it changes direction from increasing to decreasing or vice versa. These are useful for identifying local maxima or minima. In our context, we are particularly interested in where such changes occur.
A critical number is a point \( x \) where the derivative \( f'(x) \) is zero or undefined. In the example, we discovered \( x = \frac{1}{\ln(4)} \) as our critical number where \( f'(x) = 0 \).
By examining intervals to the left and right of this number, we determine if the function switches from increasing to decreasing or vice versa. At \( x = \frac{1}{\ln(4)} \), \( f(x) \) transitions from increasing to decreasing. Therefore, this critical point is a relative maximum. Understanding relative extrema helps in sketching the graph and in applications.

Derivative Calculation

Calculating derivatives is a fundamental process for analyzing functions in calculus. The derivative of a function at a point gives the slope of the tangent line to the curve at that point, which helps in understanding various behaviors of the function.
For \( f(x) = x - \log_{4} x \), the derivative \( f'(x) \) was calculated as follows:

• The derivative of \( x \) is simply \( 1 \), because \( x \) is a linear function.
• For the \( -\log_{4} x \) term, we use rules involving logarithms. Specifically, \( \frac{d}{dx}\log_{b} x = \frac{1}{x \ln(b)} \). Hence, the derivative here is \(-\frac{1}{x \ln(4)}\).

Combining these, we get \( f'(x) = 1 - \frac{1}{x \ln(4)} \).
This derivative helps us find critical numbers and thus plays a crucial role in discovering intervals of increase and decrease, as well as identifying potential relative extrema.