Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(11-18\), use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ f(x)=(x-1)^{2} $$

Step-by-Step Solution

Answer

The function is concave up for all \( x \) and has no inflection points.

Step 1: Find the First Derivative

To use the concavity theorem, we need the second derivative of the given function. First, find the first derivative of the function \( f(x) = (x-1)^2 \). \[ f'(x) = \frac{d}{dx}((x-1)^2) = 2(x-1) \]

Step 2: Find the Second Derivative

Now compute the second derivative of the function from the first derivative. \[ f''(x) = \frac{d}{dx}(2(x-1)) = 2 \]

Step 3: Analyze Concavity from the Second Derivative

The second derivative \( f''(x) = 2 \) is a constant positive number. Since \( f''(x) > 0 \) for all \( x \), according to the Concavity Theorem, the function \( f(x) = (x-1)^2 \) is concave up for all values of \( x \).

Step 4: Determine the Inflection Points

Inflection points occur where the concavity changes; this requires \( f''(x) = 0 \) or \( f''(x) \) does not exist. Since \( f''(x) = 2 \) is never zero or undefined, there are no inflection points for \( f(x) = (x-1)^2 \).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with finding properties of functions and shapes through the use of derivatives and integrals. It allows us to understand how functions change and provides tools for analyzing and predicting their behavior.
One of its key tools is the derivative, which measures the rate of change of a function. By calculating derivatives, we can find slopes of tangent lines, determine increasing or decreasing behaviors, and analyze concavity—all of which are crucial for understanding the graphing of functions.

Inflection Points

Inflection points are specific points on the graph of a function where the concavity changes direction.
In simple terms, it's where a curve switches from looking "scooping up" (concave up) to "scooping down" (concave down) or vice versa.
For a point to be classified as an inflection point, the second derivative \(f''(x)\) of the function at that point should be either zero or undefined. However, just having the second derivative equal to zero or undefined is not sufficient; the actual change in concavity around that point must be checked.
In the problem you've worked on, the function \(f(x) = (x-1)^2\) maintained the same concavity across all points, so there weren't any specific inflection points.

Concave Up

When a function is said to be "concave up," it means that its graph seems to form a bowl facing upwards.
Mathematically, this occurs when the second derivative \(f''(x)\) is positive over an interval. At these points, the slope of the first derivative is increasing.
This form of the curve intuitively indicates that as you move along the graph, the function is opening upwards, like a smile.
In the given exercise, \(f(x) = (x-1)^2\) is concave up for all values of \(x\) because the second derivative is constantly positive ( \(f''(x) = 2\)), ensuring the function never stops "smiling."

Second Derivative

The second derivative of a function, denoted as \(f''(x)\), describes how the rate of change of the rate of change is behaving. Essentially, it's the derivative of the derivative.
This is crucial because it gives insights into the concavity of a function.

• If \(f''(x) > 0\), the function is concave up.
• If \(f''(x) < 0\), the function is concave down.

Calculating the second derivative involves taking the derivative of the first derivative. In the example, the given function \(f(x) = (x-1)^2\) had a first derivative of \(2(x-1)\), and further differentiating that gives a second derivative of \(2\), which is a constant positive number, confirming the graph's upward concavity across its entire domain.