Problem 29 - Calculus for Biology and Medicine

Question

Determine all inflection points. \(f(x)=x \ln x, x>0\)

Step-by-Step Solution

Answer

The function has no inflection points.

Step 1: Find the First Derivative

The first step in finding inflection points is to determine the first derivative of the function. Let \( f(x) = x \ln x \). Using the product rule, \( (uv)' = u'v + uv' \), let \( u = x \) and \( v = \ln x \). Therefore, \( u' = 1 \) and \( v' = \frac{1}{x} \). The first derivative, \( f'(x) \), is calculated as follows: \[ f'(x) = 1 \ln x + x \frac{1}{x} = \ln x + 1. \]

Step 2: Find the Second Derivative

The second step is to find the second derivative, as inflection points occur where the concavity changes. Differentiate \( f'(x) = \ln x + 1 \) to find \( f''(x) \). Since the derivative of \( \ln x \) is \( \frac{1}{x} \) and the derivative of \( 1 \) is 0, we get: \[ f''(x) = \frac{1}{x}. \]

Step 3: Set the Second Derivative to Zero and Solve

To find possible inflection points, we set the second derivative equal to zero and solve for \( x \): \[ \frac{1}{x} = 0. \]However, \( \frac{1}{x} = 0 \) can never be true for any real \( x>0 \). This means there are no points where the second derivative changes sign. Thus, \( f(x) = x \ln x \) has no inflection points.

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change. It is divided into two main parts: differential and integral calculus. In the context of finding inflection points, differential calculus is particularly relevant. This involves calculating derivatives, which are used to determine how a function behaves at different points. By understanding the rate at which a function's value changes, we can uncover important features about the function's graph, such as slopes and curvatures. Calculus helps us identify where a function transitions between being concave up and concave down by examining these changes.

Derivatives

Derivatives are central to calculus and are crucial in understanding functions deeply. A derivative essentially represents a function's rate of change at any given point.
It gives us the slope of the tangent line to the function's curve at that point. For instance, when determining the first derivative of a function like \(f(x) = x \ln x\), we apply the product rule, which is useful for cases where functions are multiplied together:

• The product rule states \((uv)' = u'v + uv'\).
• For \(f(x) = x \ln x\), we let \(u = x\) and \(v = \ln x\).
• So, \(u' = 1\) and \(v' = \frac{1}{x}\).
• Therefore, the first derivative is \(f'(x) = \ln x + 1\).

Finding this first derivative is a necessary step in understanding the function's basic trends and how it progresses over its domain.

Concavity

Concavity pertains to the direction in which a function curves. It is an important concept that helps us understand the overall shape of a graph. To analyze concavity, we look at the second derivative of a function. The second derivative lets us know whether a function is curving upwards or downwards:

• If \(f''(x) > 0\), the function is concave up, indicating a U-shaped curve.
• If \(f''(x) < 0\), the function is concave down, suggesting an n-shaped curve.

Inflection points occur where \(f''(x) = 0\) or where \(f''(x)\) changes sign, representing a shift between concave up and concave down. However, as with \(f(x) = x \ln x\), where \(f''(x) = \frac{1}{x}\) never equals zero for \(x>0\), there are no points of sign change. Thus, this particular function does not have inflection points within its domain, meaning its concavity does not change.