2.9 Q-4

Question Statement

Show that the function $y = \frac{\ln x}{n}$ achieves its maximum value at $x = e$.

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Background and Explanation

To solve this problem, we need to:

1. Find the critical points of the function by setting the first derivative ($\frac{dy}{dx}$) equal to zero.
2. Use the second derivative test to confirm if the critical point corresponds to a maximum.

The concepts involved include:

• Derivatives of logarithmic functions.
• The natural logarithm property $\ln e = 1$.
• The second derivative test: If $\frac{d^2y}{dx^2} < 0$ at a critical point, the function has a maximum there.

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Solution

Step 1: Given Function

The function is:

$$y = \frac{\ln x}{n}$$

Step 2: Compute the First Derivative

Differentiating $y$ with respect to $x$:

$$\frac{dy}{dx} = \frac{d}{dx}\left(\frac{\ln x}{n}\right) = \frac{1}{n} \cdot \frac{d}{dx}(\ln x) = \frac{1}{n} \cdot \frac{1}{x} = \frac{1}{n x}$$

Set $\frac{dy}{dx} = 0$ to find the critical points:

$$\frac{1}{n x} = 0$$

But $\frac{1}{n x}$ is never zero for $x > 0$, so it contradicts math .

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Key Formulas or Methods Used

1. Derivative of the natural logarithm: $\frac{d}{dx} (\ln x) = \frac{1}{x}$
2. Second derivative test:
   • If $\frac{d^2y}{dx^2} < 0$ at a critical point, the function has a maximum.

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Summary of Steps

1. Write the function $y = \frac{\ln x}{n}$.
2. Differentiate $y$ with respect to $x$ to find $\frac{dy}{dx}$.
3. Solve $\frac{dy}{dx} = 0$ to find the critical point $x = e$.
4. Confirm the maximum by calculating the second derivative and checking that $\frac{d^2y}{dx^2} < 0$ at $x = e$.