Question

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ g(x)=x \sqrt{x+3} $$

Short answer

The relative extrema and points of inflection can be identified using a combination of calculus - primarily the first and second derivative tests - and verification using graphical representation of the function.

Step-by-step solution

Understand the function

First, examine the function \(g(x) = x\sqrt{x+3}\). The standard form of a function does not reveal any evident extrema or inflection points.

Compute the derivative of the function

Apply the product rule for differentiation which states if a function g(x) is the product of two other functions, say u(x) and v(x), the derivative is given by g'(x) = u'(x)v(x) + u(x)v'(x). In this case, let \(u(x) = x\) and \(v(x) = \sqrt{x+3}\). The derivative is: \(g'(x) = 1 * \sqrt{x+3} + x * (1/2\sqrt{x+3}) = \sqrt{x+3} + \frac{x}{2\sqrt{x+3}}\).

Find the critical points

Critical points occur where the derivative equals zero or is undefined. Solving \(g'(x) = 0\) yields \(x = -3\) and \(x = -1/2\). The derivative is undefined at \(x = -3\). Thus, the critical points are \(x = -3\) and \(x = -1/2\).

Apply the second derivative test

Compute the second derivative \(g''(x)\) by differentiating \(g'(x)\) and examine its sign around the critical points to identify the points of inflection and to determine where the function is concave up and concave down. Then further differentiate the function \(g'(x)\) to find \(g''(x)\).

Identify extrema and inflection points

Applying the second derivative test, identify whether each critical point is a relative maximum, a relative minimum, or a point of inflection. An extremum occurs if the sign of the second derivative changes before and after the point. If the sign doesn't change, then the critical point is an inflection point.

Use a graphing utility

Finally, graph the function using a graphing utility to validate the identified extrema and inflection points.

Key concepts

Critical Points

Critical points are essential in understanding the behavior of a function. They occur where the first derivative of a function is either zero or undefined. For our function, \(g(x) = x \sqrt{x+3}\), we derived the first derivative as \(g'(x) = \sqrt{x+3} + \frac{x}{2\sqrt{x+3}}\). To find the critical points:

• Solve \(g'(x) = 0\).
• Identify where \(g'(x)\) is undefined.

In this case, solving the equation \(g'(x) = 0\) gives us the critical points at \(x = -3\) and \(x = -1/2\). The derivative is also undefined when \(x=-3\). These points help predict where the function might have a local maximum or minimum.

Derivatives

Derivatives are foundational in calculus for understanding how a function changes. When dealing with a product of functions, the product rule is vital. The product rule states:\[ g'(x) = u'(x)v(x) + u(x)v'(x) \]where \(u(x)\) and \(v(x)\) are functions of \(x\). For function \(g(x) = x \sqrt{x+3}\), let \(u(x) = x\) and \(v(x) = \sqrt{x+3}\). Applying the derivative:

• \(u'(x) = 1\)
• \(v'(x) = \frac{1}{2\sqrt{x+3}}\)

Hence, the derivative is calculated as:\[ g'(x) = 1 \cdot \sqrt{x+3} + x \cdot \frac{1}{2\sqrt{x+3}} = \sqrt{x+3} + \frac{x}{2\sqrt{x+3}} \]Derivatives give invaluable insights like slopes and rates of change which are critical in determining the behavior of the function.

Extrema

Extrema refer to the minimum and maximum points on a function's graph. Identifying these points helps us understand the highest and lowest outputs that a function can yield within a specific range. Once we determine the critical points, we use the second derivative test to differentiate between relative maxima and minima:

• If \(g''(x) > 0\) at a critical point, it's a local minimum.
• If \(g''(x) < 0\) at a critical point, it's a local maximum.

In the function \(g(x) = x \sqrt{x+3}\), after interpreting the second derivative around critical points, we better understand the nature of the extrema by observing these sign changes.

Graphing Utility

Graphing utilities are powerful tools that visually demonstrate mathematical concepts. They allow students to see the function's behavior, confirming analytical findings like extrema and inflection points. By inputting \(g(x) = x \sqrt{x+3}\) into a graphing calculator or online graphing tool, one can:

• Observe the graph's peaks and valleys, indicating extrema.
• See where the curve changes direction, suggesting inflection points.

Graphing utilities are invaluable for verifying the manual calculations and assumptions made about the function's behavior, providing a graphical representation to support analytic work.

Inflection Points

Inflection points mark where a function changes concavity—where it shifts from concave up (shaped like a cup) to concave down (shaped like a cap), or vice versa. To determine these from a second derivative \(g''(x)\), you:

• Find where \(g''(x) = 0\).
• Ensure a sign change around these points.

An inflection point doesn't necessarily have a zero value of the first derivative. Instead, it signifies where the function moves from acceleration to deceleration, visually noticeable on a graph as a change in the curve's direction. Identifying inflection points ensures a complete understanding of how the function \(g(x) = x \sqrt{x+3}\) behaves over its domain.