Question

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ f(x)=\frac{x^{2}}{x^{2}+1} $$

Short answer

The given function \(f(x)=\frac{x^{2}}{x^{2}+1}\) is concave upward on the interval \((- \infty, \infty)\).

Step-by-step solution

Compute the first derivative

Using the quotient rule, \[f'(x) = \frac{(x^2+1) \cdot 2x - x^2 \cdot 2x}{(x^2+1)^2} = \frac{2x}{(x^2+1)^2}\].

Compute the second derivative

Again applying the quotient rule, we get \[f''(x) = \frac{(x^2+1)^2 \cdot 2 - 2x \cdot 4x(x^2+1)}{(x^2+1)^4} = \frac{2}{(x^2+1)^2}\].

Find points of inflection

Set the second derivative equal to zero and solve for x: \[0 = \frac{2}{(x^2+1)^2}\]. In this case, there are no solutions because the denominator of the fraction cannot equal zero.

Checking intervals

Since there are no inflection points, we only need to check the concavity of the function on \( (-\infty, \infty) \). By picking a test point, say x=0, we can substitute into the second derivative, thus \[f''(0) = \frac{2}{(0^2+1)^2} = 2 > 0\], this proves that the function is concave upward on \( (-\infty, \infty) \).

Key concepts

Understanding Calculus in Relation to Concavity

Calculus is the branch of mathematics that helps us understand how things change. It provides tools to explore functions, their rates of change, and how they behave. When we talk about concavity in functions, we're particularly interested in how the curve of a graph bends. Concavity can tell us a lot about the nature of a function in various intervals of its domain.

Concavity occurs in two main forms:

• Concave Upward: When the graph of the function forms a shape similar to a cup or a U. This usually indicates that the second derivative of the function is positive in that interval.
• Concave Downward: Here, the graph forms an upside-down cup or an inverted U shape. This occurs when the second derivative is negative.

By examining the second derivative, we determine the intervals of concavity, allowing us to better understand and visualize the behavior of the function. With calculus, we're able to take a deeper dive into these essential characteristics of functions.

Inflection Points and What They Indicate

Inflection points are significant in a function's curve as they mark a change in concavity. In simpler terms, they are the points where the graph shifts from curving one way to curving the opposite way. At these points, the second derivative of a function equals zero or doesn't exist; however, not every zero of the second derivative is an inflection point.

To find inflection points:

• Equate the second derivative to zero.
• Solve for the variable, usually x, to find potential points where concavity changes.
• Check these points to ensure the second derivative actually changes signs around them.

In the case of the given function \(f(x) = \frac{x^2}{x^2+1}\), the second derivative did not provide real solutions that change sign. Therefore, this function does not have inflection points, indicating uniform concavity across its domain.

Utilizing the Quotient Rule in Derivative Calculations

The quotient rule in calculus is invaluable when differentiating functions that are expressed as the quotient of two other functions. It is particularly useful when both the numerator and the denominator are differentiable. This rule helps us compute the derivative of a function like \( \frac{x^2}{x^2+1} \).

The quotient rule states:\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\cdot u' - u\cdot v'}{v^2}\]where:

• \( u \) is the numerator function
• \( v \) is the denominator function
• \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \) respectively

For our exercise, applying the quotient rule allowed us to find both the first and second derivatives important for determining concavity and inflection points. It's a perfect illustration of how calculus tools are used in understanding more complex functions.