Problem 32
Question
(a) If \(b\) is a positive constant and \(x>0\), find all critical points of \(f(x)=x-b \ln x\) (b) Use the second derivative test to determine whether the function has a local maximum or local minimum at each critical point.c
Step-by-Step Solution
Verified Answer
The critical point is \( x = b \) and is a local minimum.
1Step 1: Find the First Derivative
To find the critical points of the function, we first need to find its first derivative. The function given is \( f(x) = x - b \ln x \). Apply the derivative to get \( f'(x) = 1 - \frac{b}{x} \).
2Step 2: Set the First Derivative to Zero
Critical points occur where the first derivative is zero or undefined. Set \( f'(x) = 1 - \frac{b}{x} = 0 \). Solve for \( x \): \[ 1 = \frac{b}{x} \]\[ x = b \].
3Step 3: Determine if Critical Points are in Domain
Consider the domain \( x > 0 \), and since \( b \) is a positive constant, \( x = b \) is within the domain. Hence, the only critical point is \( x = b \).
4Step 4: Find the Second Derivative
Calculate the second derivative to apply the second derivative test. The first derivative is \( f'(x) = 1 - \frac{b}{x} \), so taking the derivative again gives:\( f''(x) = \frac{b}{x^2} \).
5Step 5: Apply the Second Derivative Test
Use the second derivative test by plugging the critical point into \( f''(x) \). For \( x = b \), \( f''(b) = \frac{b}{b^2} = \frac{1}{b} \). Since \( \frac{1}{b} > 0 \), the function has a local minimum at \( x = b \).
Key Concepts
first derivativesecond derivative testlocal maximum and minimum
first derivative
When examining a function for critical points, the first step is to find its first derivative. This helps us determine where the function's slope is zero, indicating potential peaks, valleys, or points where the function might change behavior significantly.
For the function provided, \[ f(x) = x - b \ln x, \] we need to identify its rate of change by differentiating it.
The first derivative is found as follows:
Next, we set \( f'(x) \) to zero to find critical points:\[ 1 - \frac{b}{x} = 0. \] Solving for \( x \), we find \( x = b \). Hence, the critical point occurs where the first derivative is zero, specifically at \( x = b \).
Understanding these steps forms a foundation for locating critical points and understanding how changes in the function might occur.
For the function provided, \[ f(x) = x - b \ln x, \] we need to identify its rate of change by differentiating it.
The first derivative is found as follows:
- Derivative of \( x \) is 1.
- Derivative of \( -b \ln x \) is \(- \frac{b}{x} \) following the chain rule for logarithms.
Next, we set \( f'(x) \) to zero to find critical points:\[ 1 - \frac{b}{x} = 0. \] Solving for \( x \), we find \( x = b \). Hence, the critical point occurs where the first derivative is zero, specifically at \( x = b \).
Understanding these steps forms a foundation for locating critical points and understanding how changes in the function might occur.
second derivative test
To determine the nature of the critical points found using the first derivative, we use the second derivative test. This test helps us understand whether a critical point is a local maximum, local minimum, or a saddle point.
The second derivative takes the rate of change of the first derivative, essentially showing how the slope itself changes:
To apply the second derivative test, we evaluate \( f''(x) \) at the critical point \( x = b \):\[ f''(b) = \frac{b}{b^2} = \frac{1}{b}. \]
Since \( \frac{1}{b} > 0, \) this positive value indicates that \( f(x) \) is concave up at \( x = b \), classifying it as a local minimum.
The second derivative test simplifies understanding complex function behaviors and assists in visualizing function curvature at various points.
The second derivative takes the rate of change of the first derivative, essentially showing how the slope itself changes:
- For the function's first derivative, \( f'(x) = 1 - \frac{b}{x}, \) we derive again to find \( f''(x). \)
- The derivative of \(- \frac{b}{x} \) is \( \frac{b}{x^2}. \)
To apply the second derivative test, we evaluate \( f''(x) \) at the critical point \( x = b \):\[ f''(b) = \frac{b}{b^2} = \frac{1}{b}. \]
Since \( \frac{1}{b} > 0, \) this positive value indicates that \( f(x) \) is concave up at \( x = b \), classifying it as a local minimum.
The second derivative test simplifies understanding complex function behaviors and assists in visualizing function curvature at various points.
local maximum and minimum
Critical points categorized as local maxima or minima provide crucial insights into the behavior of a function.
A local maximum occurs when a point is higher than all nearby points, while a local minimum is lower than those surrounding it. These points don't necessarily represent the highest or lowest values of the entire function, but they do indicate significant changes.
After finding the critical point, which in our function \( f(x) = x - b \ln x \) was \( x = b \), we determined its nature using the second derivative test. We found that \( f''(x) = \frac{b}{x^2}, \) and when evaluated at \( x = b \), \( f''(b) = \frac{1}{b} > 0 \).
The positive second derivative meant that the function at this critical point is concave up, similar to how a smile appears.
Thus, the point \( x = b \) is a local minimum. In this context:
A local maximum occurs when a point is higher than all nearby points, while a local minimum is lower than those surrounding it. These points don't necessarily represent the highest or lowest values of the entire function, but they do indicate significant changes.
After finding the critical point, which in our function \( f(x) = x - b \ln x \) was \( x = b \), we determined its nature using the second derivative test. We found that \( f''(x) = \frac{b}{x^2}, \) and when evaluated at \( x = b \), \( f''(b) = \frac{1}{b} > 0 \).
The positive second derivative meant that the function at this critical point is concave up, similar to how a smile appears.
Thus, the point \( x = b \) is a local minimum. In this context:
- The graph of the function would dip down, reaching its lowest at \( x = b \) before ascending, illustrating a local minimum.
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