Question

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{4}+3}{x^{2}+1} $$

Short answer

If all solutions are valid, the points on the graph of the function \(f(x)\) where it has a horizontal tangent are \((x1, y1), (x2, y2), \ldots\) depending on the solutions from the derivative equation.

Step-by-step solution

Apply the derivative using the quotient rule

First, we need to differentiate the function \(f(x) = \frac{x^4 + 3}{x^2 + 1}\) using the quotient rule.

Simplify the derivative

After applying the quotient rule, simplify the derivative calculated in the previous step.

Determine when the derivative is zero

Set the derivative function equal to zero and solve for \(x\).

Validate solutions

Check if all solved values of \(x\) are within the function's domain.

Find the corresponding \(y\)-coordinates

Finally, plug the \(x\)-coordinate(s) into the original function to find the corresponding \(y\)-coordinates of the point(s) where the function has a horizontal tangent.

Key concepts

Derivative

Understanding derivatives is key in determining where a function has a horizontal tangent. The derivative of a function, often denoted as \(f'(x)\), provides the slope of the tangent line at any given point on the function's graph. When the derivative is equal to zero, the slope of the tangent line is zero. This indicates that the tangent is horizontal.
To find where the graph of \(f(x) = \frac{x^4 + 3}{x^2 + 1}\) has a horizontal tangent, we need to differentiate the function and set the derivative equal to zero. In this context, the derivative helps us understand changes in the graph as it indicates points where the function stops increasing or decreasing momentarily before potentially changing direction.

Quotient Rule

The quotient rule is vital when differentiating functions that are in the form of a fraction, such as \(\frac{x^4 + 3}{x^2 + 1}\). The rule states that if you have a function \(f(x) = \frac{g(x)}{h(x)}\), its derivative is \[f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\] Using this rule correctly is critical for computing derivatives of fractions. In our exercise, applying the quotient rule leads to a derived function which can be simplified until we solve where it equals zero. This step-by-step application ensures accuracy when attempting to find the horizontal tangents. Remember that each part of the fraction expresses its respective derivative as prescribed in the quotient rule's formula.

Domain

The domain of a function refers to all the possible input values (often represented as \(x\)-values) for which the function is defined and produces real outputs. Understanding the domain is crucial, especially when validating solutions after finding potential points of interest, like where the derivative equals zero.
For our function \(f(x) = \frac{x^4 + 3}{x^2 + 1}\), the domain is all real numbers. This is because the denominator, \(x^2 + 1\), never becomes zero as it always has a positive output. Therefore, there are no values of \(x\) that make the function undefined within the realm of real numbers. Checking that solutions fall within the domain is a necessary final step after finding \(x\)-values where the derivative is zero. This ensures all points with horizontal tangents are valid and real.

Function Graph

The function graph offers a visual representation of a function, showing the relationship between \(x\)-values and their corresponding \(y\)-values from the function. A crucial part of analyzing any function is interpreting its graph. For the function \(f(x) = \frac{x^4 + 3}{x^2 + 1}\), locating horizontal tangents involves viewing the points where the slope of the tangent line (indicated by the derivative) is zero.
Once we've calculated the \(x\)-values where \(f'(x) = 0\), plotting these on the function graph helps us see where the function's direction changes, potentially forming peaks or troughs. Plugging back these \(x\)-values into the original function gives us the exact \((x, y)\) coordinates for these horizontal tangents, allowing for precise graph interpretation and analysis. The graph is a visual tool to confirm and support mathematical findings, providing a clearer understanding of the function's behavior.