Problem 19
Question
Find the points of inflection and discuss the concavity of the graph of the function. \(y=x-\ln x\)
Step-by-Step Solution
Verified Answer
The function \(y=x-\ln x\) has no points of inflection and is concave up for all \(x>0\).
1Step 1: Find the First Derivative
The derivative of \(y=x-\ln x\) is given by \(\frac{dy}{dx}=1-\frac{1}{x}\). This is gotten by direct application of logarithmic derivative rule.
2Step 2: Find the Second Derivative
The derivative of \(1-\frac{1}{x}\) is \(\frac{dy}{dx^2}=\frac{1}{x^2}\). This is found using the power and chain rules.
3Step 3: Find Points of Inflection
To find the points of inflection, we set the second derivative to zero and solve for x, \(\frac{1}{x^2}=0\). However, there are no real solutions to this equation, meaning that there are no points of inflection.
4Step 4: Discuss the Concavity of the Graph
We know that the graph is concave up where the second derivative is positive, and concave down where the second derivative is negative. Checking the signs of the second derivative, \(\frac{1}{x^2}\), we find that it is always positive for x>0. This means our graph is always concave up for x>0.
Key Concepts
Inflection PointsConcavityDerivativesLogarithmic Functions
Inflection Points
Inflection points are special locations on a graph where the curvature changes direction. To find these, you need to look at the second derivative of a function. The place where a graph moves from concave up to concave down or vice versa, typically highlights an inflection point.
For the function in the problem, the second derivative is given as \( \frac{1}{x^2} \). To find inflection points, this derivative should be equal to zero, but this does not happen for \( x > 0 \). This indicates there are no points of inflection for \( y = x - \ln x \).
Inflection points might seem puzzling, but once you understand the role of the second derivative, they become clearer.
For the function in the problem, the second derivative is given as \( \frac{1}{x^2} \). To find inflection points, this derivative should be equal to zero, but this does not happen for \( x > 0 \). This indicates there are no points of inflection for \( y = x - \ln x \).
Inflection points might seem puzzling, but once you understand the role of the second derivative, they become clearer.
- An inflection point is where the graph changes from concave to convex or vice versa.
- For an inflection point to exist, the second derivative must be zero or undefined.
Concavity
Concavity tells us about the direction the graph curves. It is all about the "bowl" shape of the graph. If a graph is concave up, it looks like a bowl facing upwards, while concave down resembles a bowl facing downwards.
To determine the concavity of \( y = x - \ln x \), we look at the second derivative \( \frac{1}{x^2} \):
To determine the concavity of \( y = x - \ln x \), we look at the second derivative \( \frac{1}{x^2} \):
- Since \( \frac{1}{x^2} \) is positive for \( x > 0 \), the graph is always concave up in this region.
- Concave down would occur where the second derivative is negative, which does not happen here.
Derivatives
Derivatives are fundamental in calculus, allowing us to find how a function changes. The first derivative of a function provides the slope of the tangent line to the graph at any point. For the function \( y = x - \ln x \), the first derivative is \( 1 - \frac{1}{x} \). This tells us how the function's value changes as \( x \) changes.
To find how fast or in what direction \( y = x - \ln x \) is changing, use the first derivative:
To find how fast or in what direction \( y = x - \ln x \) is changing, use the first derivative:
- A positive value means the function is increasing.
- A negative value tells us the function is decreasing.
Logarithmic Functions
Logarithmic functions are an essential component in mathematics, especially in calculus. They are functions of the form \(\ln(x)\), where \(\ln\) is the natural logarithm.
In \( y = x - \ln x \), the logarithmic function part, \(-\ln x\), plays a crucial role. It influences the behavior of the function significantly, especially when considering derivative calculations. The derivative of \( \ln x \) is \( \frac{1}{x} \), which is essential for finding how the function changes, providing important insights into the overall function behavior.
In \( y = x - \ln x \), the logarithmic function part, \(-\ln x\), plays a crucial role. It influences the behavior of the function significantly, especially when considering derivative calculations. The derivative of \( \ln x \) is \( \frac{1}{x} \), which is essential for finding how the function changes, providing important insights into the overall function behavior.
- Logarithmic functions grow slower than polynomial functions.
- They appear in various real-world contexts, such as measuring growth rates.
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