Problem 65 - Calculus Single Variable

Question

Suppose that \(00\), let $$ f(x)=\frac{x^{p}}{x^{2}+1}. $$ Find the point at which the graph of \(f\) has a horizontal tangent line.

Step-by-Step Solution

Answer

The graph of \( f(x) \) has a horizontal tangent line at \( x = \sqrt{\frac{p}{2-p}} \).

Step 1: Differentiate the Function

To find where the graph has a horizontal tangent line, we first need to find its derivative and set it to zero. The derivative, \( f'(x) \), can be found using the quotient rule. If \( f(x) = \frac{x^p}{x^2 + 1} \), then by the quotient rule: \( f'(x) = \frac{(x^2 + 1) \, p \, x^{p-1} - x^p \, 2x}{(x^2 + 1)^2} \). Simplifying this gives:\[ f'(x) = \frac{p x^{p-1} (x^2 + 1) - 2x^{p+1}}{(x^2 + 1)^2} \].

Step 2: Simplify the Derivative

Simplify the numerator of \( f'(x) \): \[ f'(x) = \frac{p x^{p-1} x^2 + p x^{p-1} - 2x^{p+1}}{(x^2 + 1)^2} \]. This simplifies to: \[ f'(x) = \frac{p x^{p+1} + p x^{p-1} - 2x^{p+1}}{(x^2 + 1)^2} \] or \[ f'(x) = \frac{(p - 2)x^{p+1} + px^{p-1}}{(x^2 + 1)^2} \].

Step 3: Set the Derivative to Zero

For the tangent line to be horizontal, \( f'(x) = 0 \). Thus, we set the simplified numerator to zero:\[ (p - 2)x^{p+1} + px^{p-1} = 0. \] Factor the common term \( x^{p-1} \):\[ x^{p-1}((p - 2)x^2 + p) = 0. \] Since \( x > 0 \), \( x^{p-1} eq 0 \). Therefore, we solve:\[ (p - 2)x^2 + p = 0. \]

Step 4: Solve for x

From the equation \[ (p - 2)x^2 + p = 0, \]rearrange to find: \[ (p - 2)x^2 = -p. \] If \( p - 2 eq 0 \), divide both sides by \( p - 2 \):\[ x^2 = \frac{-p}{p - 2}. \] Taking the square root, we find:\[ x = \sqrt{\frac{p}{2-p}}. \] Since \( x > 0 \), we use the positive square root: \[ x = \sqrt{\frac{p}{2-p}}. \]

Key Concepts

Horizontal Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing over. The slope of a tangent line at a particular point on a function's graph represents the derivative of the function at that point.
When a tangent line is horizontal, it means that its slope is zero. This is of particular interest in calculus because it often indicates a local maximum or minimum point, or a point of inflection. To determine where a function has a horizontal tangent line:

• Take the derivative of the function.
• Set the derivative equal to zero and solve for the variable.

Here, finding where the graph of the function has a horizontal tangent line can give us critical insight into the behavior of the function, such as identifying peaks and valleys in its graph.

Quotient Rule

The quotient rule is a technique used for differentiating functions that are the quotient of two other functions. If you have a function defined as a fraction, it is crucial to know how to apply this rule.
For a function given by:

• \( f(x) = \frac{u(x)}{v(x)} \)

The quotient rule states that the derivative of \( f(x) \) is:

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

This formula is essential because it allows you to differentiate functions without having to simplify the quotient itself before differentiating, thereby preserving more complex expressions. Applying the quotient rule was a key step in the original exercise for finding where a horizontal tangent line exists.

Simplifying Expressions

After applying the quotient rule, it's common to end up with a fairly complicated expression. Simplifying expressions in calculus helps to make the problem more manageable and easier to solve.
During simplification, you usually:

• Combine like terms.
• Factor out common factors.
• Cancel out terms when possible.

In our exercise, simplifying involved combining terms after applying the quotient rule and factoring where applicable. This step is crucial because it reduces the complexity of the expression, making subsequent mathematical operations, like solving equations, more straightforward and efficient.

Solving Equations

Once the derivative has been found and simplified, the next step usually involves solving an equation. This is because where the derivative equals zero, the original function might have a horizontal tangent line.
In our exercise, solving the equation \((p - 2)x^2 + p = 0\) after simplification helped find the specific value of \(x\) where the function has a horizontal tangent line.
The process generally includes:

• Isolating the variable of interest.
• Applying algebraic techniques like factoring or using the quadratic formula.
• Considering any constraints given in the problem (e.g., \(x > 0\)).

During this step, real-world interpretation of solutions is often necessary to apply the mathematical solution correctly within the context provided by the problem.