Find the Maximum/Minimum Value ( natural log of x)/x

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 $\frac{d}{dx}(x^n)=n{x}^{n-1}$ 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 $n{x}^{n-1}$.

• 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.