Problem 19 - Calculus Early Transcendentals

Question

Find the intervals on which \(f\) is increasing and decreasing. Superimpose the graphs of \(f\) and \(f^{\prime}\) to verify your work. $$f(x)=(x-1)^{2}$$

Step-by-Step Solution

Answer

Answer: The function \(f(x) = (x-1)^2\) is decreasing on the interval \((-\infty, 1)\) and increasing on the interval \((1, \infty)\).

Step 1: Find the first derivative of \(f(x)\).

We start by finding the derivative of \(f(x) = (x-1)^2\). We'll apply the power rule for derivatives: \((x^n)' = nx^{n-1}\). In this case, n = 2. $$f'(x) = 2(x-1)^{2-1} \cdot \frac{d}{dx}(x-1)$$ The derivative of the inner function (x-1) is 1, so: $$f'(x) = 2(x-1)^1$$ $$f'(x) = 2(x-1)$$

Step 2: Determine the intervals of increasing and decreasing.

We'll now figure out where the first derivative is positive (increasing) and negative (decreasing). To do this, we'll set \(f'(x)\) equal to zero and solve for x: $$2(x-1) = 0$$ $$x-1 = 0$$ $$x = 1$$ Now we'll pick test points in the intervals \((-\infty, 1)\) and \((1, \infty)\) to check whether \(f'(x)\) is negative or positive: For x < 1 (e.g. x=0): \(f'(0) = 2(0-1) = -2 < 0\). This means that the function is decreasing in the interval \((-\infty, 1)\). For x > 1 (e.g. x=2): \(f'(2) = 2(2-1) = 2 > 0\). This means that the function is increasing in the interval \((1, \infty)\).

Step 3: Superimpose the graphs of \(f(x)\) and \(f'(x)\).

To verify our work, we can graph both \(f(x) = (x-1)^2\) and \(f'(x) = 2(x-1)\). It should be observable from the graphs that when \(f'(x) > 0\) (above x-axis), \(f(x)\) is increasing, and when \(f'(x) < 0\) (below x-axis), \(f(x)\) is decreasing. You can use a graphing tool to superimpose the graphs of both functions or sketch them on paper. Inspect the graphs for increasing and decreasing intervals to verify our previous calculations. In conclusion, the function \(f(x)=(x-1)^2\) is decreasing on the interval \((-\infty, 1)\) and increasing on the interval \((1, \infty)\).

Key Concepts

First Derivative Test

To determine where a function is increasing or decreasing, calculus students often use the first derivative test. By deriving the function and then analyzing the sign of the resulting first derivative, you can discover these critical intervals.

When the first derivative of a function, denoted as \( f'(x) \), is positive, the function \( f(x) \) is increasing. Conversely, if \( f'(x) \) is negative, \( f(x) \) is decreasing. The point where \( f'(x) \) changes sign is known as a critical point and can indicate a local maximum or minimum.

Specifically, to apply the first derivative test, follow these steps:

• Find the derivative of the function, \( f'(x) \).
• Calculate the critical points by setting \( f'(x) = 0 \) and solving for \( x \).
• Pick test points from the intervals divided by the critical points and determine the sign of \( f'(x) \).
• Analyze the sign changes to identify the intervals where the function increases or decreases.

In our exercise, once we found the derivative \( f'(x) = 2(x-1) \), we saw that the critical point is at \( x = 1 \). By testing values on either side of this critical point, we confirmed the function is decreasing when \( x < 1 \) and increasing when \( x > 1 \).

Power Rule for Derivatives

A fundamental tool in calculus when encountering polynomial functions is the power rule for derivatives. This rule simplifies finding a derivative, especially when you're dealing with exponents. In general, the power rule states that for any term with \( x^n \), the derivative is \( nx^{n-1} \).

Applying the power rule to our example function \( f(x) = (x-1)^2 \), the exponent in this case is 2, so the derivative is \( 2(x-1)^{2-1} \) or simplifying \( 2(x-1) \). This step remains essential in identifying the behavior of the function across different intervals.

Remember that for more complex functions, like those involving a composition of functions (a function within a function), you'll also need the chain rule in conjunction with the power rule to find the correct derivative.

Graphical Verification of Increasing and Decreasing Functions

Graphical verification is a valuable method to confirm that our calculus computations align with the actual behavior of the function. By plotting the graphs of \( f(x) \) and its derivative \( f'(x) \), we add a visual layer to our understanding of the function's increase and decrease patterns.

The steps to graphically verify increasing and decreasing intervals are:

• Graph both the original function and its derivative on the same axes.
• Observe the derivative graph: Where it's above the x-axis, the original function should be increasing, while where it's below, the function should be decreasing.
• Notice that the zeroes of \( f'(x) \) typically line up with either peaks or troughs (maxima or minima) on the \( f(x) \) graph.

By aligning our graphs from the exercise, we notice the section of the curve \( f(x) = (x-1)^2 \) slopes upward to the right of \( x = 1 \), confirming the increasing interval. To the left, the function slopes downward, validating the decreasing interval \( (-fty, 1) \). These graphical observations complement and reinforce the findings from the first derivative test.