Find the Maximum/Minimum Value ( natural log of x)/x
This is an optimization problem in calculus. The question is asking you to determine the highest or lowest possible value of the function \(f(x) = \frac{\ln(x)}{x}\) for \(x\) in its domain. You will need to use calculus techniques, such as finding the first derivative \(f'(x)\), setting it equal to zero to find critical points, and possibly examining the second derivative \(f''(x)\) to determine whether these critical points correspond to a local maximum, a local minimum, or neither. Depending on the context, you may also be asked to consider the endpoints of the function's domain or the behavior as \(x\) approaches infinity or zero, if these values are within the domain of the function.
$\frac{ln \left(\right. x \left.\right)}{x}$
Answer
Solution:
Step 1: Derive the first derivative of the function.
Step 1.1: Apply the Quotient Rule for differentiation: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{(g(x))^2}$, where $f(x) = \ln(x)$ and $g(x) = x$.
Step 1.2: The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$.
Step 1.3: Utilize the Power Rule for differentiation.
Step 1.3.1: Simplify the fraction by combining $x$ and $\frac{1}{x}$.
Step 1.3.2: Cancel out the common $x$ factor.
Step 1.3.2.1: Perform the cancellation.
Step 1.3.2.2: Rewrite the simplified expression.
Step 1.3.3: Apply the Power Rule, where $\frac{d}{dx}(x^n) = nx^{n-1}$ and $n=1$.
Step 1.3.4: Multiply $-1$ by $1$ to get the first derivative: $\frac{1 - \ln(x)}{x^2}$.
Step 2: Derive the second derivative of the function.
Step 2.1: Use the Quotient Rule again, with $f(x) = 1 - \ln(x)$ and $g(x) = x^2$.
Step 2.2: Differentiate the function.
Step 2.2.1: Raise $(x^2)^2$ to the power of $4$.
Step 2.2.1.1: Apply the rule $(a^m)^n = a^{mn}$.
Step 2.2.1.2: Multiply the exponents.
Step 2.2.2: Apply the Sum Rule for the derivative of $1 - \ln(x)$.
Step 2.2.3: Since $1$ is a constant, its derivative is $0$.
Step 2.2.4: Add $0$ and the derivative of $-\ln(x)$.
Step 2.2.5: The derivative of $-\ln(x)$ is $-\frac{1}{x}$.
Step 2.3: The derivative of $\ln(x)$ is $\frac{1}{x}$.
Step 2.4: Apply the Power Rule.
Step 2.4.1: Combine $x^2$ and $\frac{1}{x}$.
Step 2.4.2: Cancel out the common $x$ factor.
Step 2.4.2.1: Factor out $x$.
Step 2.4.2.2: Perform the cancellation.
Step 2.4.3: Apply the Power Rule with $n=2$.
Step 2.4.4: Simplify by factoring out $x$.
Step 2.5: Cancel out common factors.
Step 2.6: Simplify the expression.
Step 2.6.1: Apply the distributive property.
Step 2.6.2: Simplify the numerator.
Step 2.6.3: Factor out $-1$.
Step 3: Find critical points by setting the first derivative to zero.
- Step 3.1: Set $\frac{1 - \ln(x)}{x^2} = 0$.
Step 4: Solve for $x$ in the critical point equation.
Step 4.1: Set the numerator equal to zero: $1 - \ln(x) = 0$.
Step 4.2: Solve for $x$: $x = e$.
Step 5: Determine if the critical point is a maximum or minimum.
Step 5.1: Evaluate the second derivative at $x = e$.
Step 5.2: If the second derivative is negative, $x = e$ is a local maximum.
Step 6: Find the y-value at the critical point.
Step 6.1: Substitute $x = e$ into the original function: $f(e) = \frac{\ln(e)}{e}$.
Step 6.2: Since $\ln(e) = 1$, the y-value is $\frac{1}{e}$.
Step 7: Conclude the local extrema.
- Step 7.1: The local maximum is at $(e, \frac{1}{e})$.
Knowledge Notes:
The Quotient Rule is used for differentiating ratios of functions.
The Power Rule states that the derivative of $x^n$ is $nx^{n-1}$.
The derivative of $\ln(x)$ is $\frac{1}{x}$.
The second derivative test determines whether a critical point is a maximum or minimum based on the sign of the second derivative.
Logarithmic properties are used to simplify expressions involving logarithms.
The natural logarithm function, $\ln(x)$, has a base of $e$, where $e$ is the Euler's number, approximately equal to 2.71828.
