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A horizontal tangent line to a curve occurs when the slope of the tangent is zero. In differential calculus, this corresponds to setting the derivative of a function to zero and solving for the variable. For the given function, \( f(x) = \frac{\ln x}{x} \), to find horizontal tangents, calculate the derivative \( f'(x) = \frac{1 - \ln x}{x^2} \). This derivative represents the slope at any point on the curve.
By solving the equation \( f'(x) = 0 \), you find \( 1 - \ln x = 0 \) which simplifies to \( \ln x = 1 \). The solution to this is \( x = e \), which is approximately 2.718. Thus, the only horizontal tangent for this function occurs precisely at \( x = e \). Here, the curve of the function has a "flat" spot, meaning the slope, both left and right of this point, nearly matches before increasing or decreasing again.

The quotient rule is a technique for differentiating functions that are in the form of a fraction, \( \frac{u}{v} \). It states:

• Find the derivative of the numerator \( u \), denoted as \( u' \).
• Find the derivative of the denominator \( v \), denoted as \( v' \).
• Apply the formula for the derivative: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]

In our function \( f(x) = \frac{\ln x}{x} \), \( u = \ln x \) and \( v = x \). The derivatives are \( u' = \frac{1}{x} \) and \( v' = 1 \). Plug these into the formula to get the function's derivative: \( f'(x) = \frac{\left( \frac{1}{x} \right)x - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \).This formula helps us determine critical points where the derivative equals zero or does not exist, which can reveal horizontal or vertical tangents or other function features.

Understanding the properties of functions is crucial in calculus. The function \( f(x) = \frac{\ln x}{x} \) has distinct characteristics that determine various behaviors on its graph.

• **Intersection with the x-axis:**To see where \( f(x) \) crosses the x-axis, set the function equal to zero: \( \frac{\ln x}{x} = 0 \). This simplifies to \( \ln x = 0 \), giving \( x = 1 \). Thus, the function intersects the x-axis only at \( x = 1 \).
• **Is it many-one?**A many-one function can map multiple inputs to the same output. Since the derivative of the function \( f'(x) \) shows a changing sign around \( x = e \) which indicates a decreasing pattern after that point, it's not many-one. Each \( x \) has a unique output.
• **Vertical Tangents:**For vertical tangents, the derivative must approach infinity as the x-value approaches a certain point. Examining \( f'(x) = \frac{1 - \ln x}{x^2} \), as \( x \to 0^+ \), \( \ln x \) tends to \(-\infty \), thus not producing a valid vertical tangent. As \( x \to \infty \), the derivative tends to zero, ensuring no vertical tangents exist for the function.

These properties make up the unique characteristics of \( f(x) \) and inform us about its graph's shape and structure.