Question

Find the \(x\) -values where the graph of the function has a horizontal tangent line. $$f(x)=\frac{x}{x+1}$$

Short answer

The graph does not have any horizontal tangent lines.

Step-by-step solution

Understand the Problem

To find where the graph of the function has a horizontal tangent line, we need to determine where the derivative \(f'(x)\) is equal to zero.

Find the Derivative of the Function

First, find the derivative of the function \(f(x) = \frac{x}{x+1}\) using the quotient rule: \[ f'(x) = \frac{(x+1) \cdot 1 - x \cdot 1}{(x+1)^2} = \frac{x+1-x}{(x+1)^2} = \frac{1}{(x+1)^2} \]

Set the Derivative to Zero

For a horizontal tangent line, set the derivative equal to zero:\[ \frac{1}{(x+1)^2} = 0 \] Since there are no real values of \(x\) that satisfy this equation, the derivative \(\frac{1}{(x+1)^2}\) cannot be zero.

Conclusion

Since the derivative of the function \(f(x) = \frac{x}{x+1}\) is never zero for any real \(x\), the graph of this function does not have any horizontal tangent lines.

Key concepts

Understanding Derivatives

A derivative represents the rate at which a function changes at any given point along its curve. It's a crucial tool in calculus for finding slopes of lines tangent to curves at specific points. When you find the derivative of a function, you're essentially finding a new function that gives you the slope of the original function at any point.
For example, if you have a function like \(f(x) = \frac{x}{x+1}\), the derivative \(f'(x)\) will tell you how steep the curve is at any value of \(x\). If you want to find when the tangent is horizontal, you need to set this derivative equal to zero, because a horizontal line has a slope of zero.
In the context of the exercise, our goal was to find where \(f'(x) = 0\) for the function \(f(x) = \frac{x}{x+1}\). However, as we saw, the derivative \(\frac{1}{(x+1)^2}\) never equals zero, indicating there are no horizontal tangents.

Applying the Quotient Rule

The quotient rule is a method for taking the derivative of a function that is the ratio of two other functions. Let's say your function is \(f(x) = \frac{u(x)}{v(x)}\), which means you have \(u(x)\) as the numerator and \(v(x)\) as the denominator.
The rule states that the derivative is given by:

• \(f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}\)

Here, \(u'(x)\) is the derivative of the top function and \(v'(x)\) is the derivative of the bottom function. For \(f(x) = \frac{x}{x+1}\), we find:

• The derivative of \(x\) is 1.
• The derivative of \(x+1\) is also 1.

Substituting these into the quotient rule gives us \(f'(x) = \frac{(x+1)(1) - x(1)}{(x+1)^2}\). Simplifying this expression results in \(\frac{1}{(x+1)^2}\).
Notice that in this specific example, the derivative cannot be zero, meaning there are no horizontal tangent lines on the function graph.

Interpreting the Function Graph

Function graphs are a visual representation of how the function behaves across different values of \(x\). They allow you to see where the function increases, decreases, or stays constant.
A horizontal tangent line occurs at a point on the graph where the curve switches direction from increasing or decreasing, essentially appearing flat. This is why a tangent line with a slope of zero (derived from a derivative value of zero) is crucial for identifying such points.
For our function \(f(x) = \frac{x}{x+1}\), since the derivative \(\frac{1}{(x+1)^2}\) cannot be zero, the graph of the function never has a point where the tangent line is horizontal.
Understanding the behavior of the derivative and its zeros can tell you a great deal about the shape and trajectory of the function graph itself.