Question

In Exercises, use a graphing utility to graph \(f, f^{\prime}\). and \(f^{\prime \prime}\) in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of \(f\). State the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) $$ f(x)=\frac{x^{2}}{x^{2}+1}, \quad[-3,3] $$

Short answer

The first derivative \(f'(x) = \frac{2x(x^2+1) - 2x^3}{(x^2+1)^2}\) and the second derivative \(f''(x)=\frac{-4x^3}{(x^2+1)^3} + \frac{6x}{(x^2+1)^2}\). The behavior of \(f(x)\) (relative extrema and concavity) aligns with the signs of \(f'(x)\) and \(f''(x)\).

Step-by-step solution

Find the First Derivative \(f'\)

The first derivative of a function can be found using simple differentiation rules. The derivative of \(f(x)=\frac{x^{2}}{x^{2}+1}\) is \(f'(x) = \frac{2x(x^2+1) - 2x^3}{(x^2+1)^2}\).

Find the Second Derivative \(f''\)

The second derivative of a function can be found by differentiating the first derivative. Differentiating \(f'(x)\) yields \(f''(x)=\frac{-4x^3}{(x^2+1)^3} + \frac{6x}{(x^2+1)^2}\).

Plot \(f\), \(f'\), and \(f''\)

Plot the three functions \(f(x)\), \(f'(x)\), and \(f''(x)\) on a graph for \(x\) in the interval \([-3,3]\). Notice where the graph of \(f(x)\) has local maximums or minimums (where \(f'(x)\) crosses from positive to negative or vice versa), and where it changes concavity (where \(f''(x)\) crosses from positive to negative or vice versa).

State the relationship

Where the graph of \(f(x)\) is increasing, \(f'(x)\) is positive, and where it's decreasing, \(f'(x)\) is negative. When the graph of \(f(x)\) is concave up, \(f''(x)\) is positive, and when it's concave down, \(f''(x)\) is negative. This shows a correlation between the signs of the derivatives and the behavior of the graph of \(f(x)\).

Key concepts

Understanding the First Derivative

The first derivative of a function, commonly represented as \( f'(x) \), is crucial in analyzing the function's behavior. To find it, you apply differentiation rules to the original function. For example, in our exercise, the function \( f(x)=\frac{x^{2}}{x^{2}+1} \) has a first derivative \( f'(x) = \frac{2x(x^2+1) - x^3}{(x^2+1)^2} \). This derivative holds the keys to understanding the speed and direction of \( f(x) \) as \( x \) changes.

The first derivative helps us determine where the function is increasing or decreasing. Specifically, if \( f'(x) > 0 \), the function is increasing at that point. If \( f'(x) < 0 \), it's decreasing. This information is incredibly useful when trying to identify relative extrema—peaks and troughs on the graph of \( f(x) \).

To ensure the content is comprehensible, visualize the process: when the slope (first derivative) transitions from positive to negative, we've found a hilltop, a local maximum. Conversely, when the slope changes from negative to positive, we've found a valley, a local minimum. Understanding the behavior of the first derivative is pivotal in mapping the journey of our function over its domain.

Delving into the Second Derivative

The second derivative, \( f''(x) \), is derived from the first derivative and provides insights into the function's concavity—how it curves. In our example, calculating \( f''(x) \) from \( f'(x) \) yields \( f''(x)=\frac{-4x^3}{(x^2+1)^3} + \frac{6x}{(x^2+1)^2} \).

Understanding the second derivative is essential for identifying points of inflection, locales where the function's curve changes direction. If \( f''(x) > 0 \), the graph is concave up like a cup that could hold water. If \( f''(x) < 0 \), it's concave down like an arch.

Breaking it down further:

• Concave Up: \( f''(x) > 0 \) — the graph forms a 'U' shape.
• Concave Down: \( f''(x) < 0 \) — the graph forms an 'n' shape.

The points of inflection occur where \( f''(x) = 0 \) or is undefined, indicating a shift from concave up to concave down or vice versa. Recognize that these points are like the twists and turns in the roller coaster of the graph, revealing where it's about to go in a new direction.

Finding Relative Extrema

Relative extrema are the highs and lows within a specific range of a function’s domain. They are considered 'local' because they represent the maximum or minimum points when compared to nearby points, not the entire graph. To find relative extrema, we must look at the first derivative and find where it is zero: these are our critical points.

However, not all critical points are relative extrema. We also use the first derivative test, examining whether the sign of \( f'(x) \) changes at these critical points:

• If \( f'(x) \) changes from positive to negative, there's a relative maximum.
• If \( f'(x) \) changes from negative to positive, there's a relative minimum.

In the exercise, we graph \( f'(x) \) alongside \( f(x) \) to visually confirm these points. By carefully studying these changes in the graph, you can pinpoint where the function reaches its peak or dips into a valley, providing essential information about the function's overall behavior.

Points of Inflection and Their Significance

Points of inflection are captivating features on a graph where a function's concavity switches. These are the inflection points—where the function’s curve changes from bending one way to another.

To identify them, we scrutinize the second derivative, \( f''(x) \), seeking points where it is zero or does not exist. The existence of points of inflection doesn't just tell a story about the shape of the graph; they offer deep insights into the behavior of the function and its derivative. A point of inflection is where the acceleration of the function—how quickly it's speeding up or slowing down—shifts.

By plotting \( f(x), f'(x), \) and \( f''(x) \) on the same graph, you can visually match up these points of inflection with changes on the graph of \( f(x) \). This graphing exercise solidifies comprehension, illustrating how a function can go from being shaped like a smile to resembling a frown, or vice versa, indicating the nuanced inflection points.