Question

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ g(x)=(x-6)(x+2)^{3} $$

Short answer

The relative extrema and points of inflection of the function \( g(x) \) are identified after finding the first and second derivatives and setting them to zero, then testing these values. The function is then plotted using a graphing utility and the relative extrema and points of inflection are indicated on the graph.

Step-by-step solution

Find the First Derivative

First, find the derivative of the function \( g(x) \), which is denoted \( g'(x) \). This will be used to find the relative extrema. Use the product rule ( \( (uv)' = u'v + uv' \) ) for differentiation to find the first derivative of the function \( g(x) \). After differentiating the given function \( g(x)=(x-6)(x+2)^{3} \), we receive \( g'(x) = 3(x+2)^2 + (x-6)(3)(x+2)^2 \). Now, in order to identify the relative extrema, we need to find the values of \( x \) where \( g'(x) = 0 \). Solve the equation \( g'(x) = 0 \) for \( x \).

Identify Relative Extrema

Substitute the values found from Step 1 into the original function \( g(x) \) to determine whether they are relative maximums, relative minimums, or neither. For each \( x \) value, look at function values for slightly larger and slightly smaller \( x \)-values. If the \( g(x) \) value is the highest among its neighbors, it is a relative maximum; if it's the lowest, it is a relative minimum.

Find the Second Derivative

The second derivative, denoted \( g''(x) \), can be found by once again applying the product and chain rules of differentiation on \( g'(x) \). The second derivative will be involved in finding the points of inflection.

Identify Points of Inflection

A point of inflection is a place where the function changes concavity. This occurs when the second derivative, \( g''(x) \), is equal to zero or undefined. So, set \( g''(x) \) to zero and solve for \( x \) to find possible points of inflection. Then, test these points by calculating the values of the second derivative at points slightly less than and slightly more than each possible point of inflection. If the sign of \( g''(x) \) changes at the point, it's a point of inflection.

Graph the Function

Now that we know the relative extrema and points of inflection, plot the points on a graph and then sketch the graph of \( g(x) \) using a graphing utility.

Key concepts

Graphing Utility

A graphing utility is a helpful tool in visualizing mathematical functions. It can be a software or a device like a graphing calculator.
They allow you to input equations and see their corresponding graphs. For functions that involve variables like in calculus, using a graphing utility provides an immediate visual aid.
It's particularly useful for identifying key features within graph functions, such as turning points and changes in concavity.
Here’s how a graphing utility can help you in calculus:

• Accurate plotting: Ensures that the function is accurately represented.
• Dynamic visualization: Helps in understanding how the function behaves by manipulating variables.
• Finding extrema and inflection points: Makes it easier to identify these important points through graphical inspection.

Using a graphing utility can significantly simplify complex analysis, aiding in both learning and verifying your calculations.

Relative Extrema

Relative extrema refer to the high and low points on a graph within a given interval. These are either relative maxima or minima and occur where the slope of the tangent line is zero, meaning the first derivative is zero.
To identify these points, follow these steps:

• Find the first derivative of the function, which provides a formula for its slope.
• Set the first derivative equal to zero and solve for the variable to find potential extrema points.
• Examine the function at these points, considering nearby values to decide if each is a max or min.

Once the relative extrema are identified, you get a clearer understanding of the function’s peaks and valleys, which is critical for analyzing behaviors like optimization problems in calculus.

Points of Inflection

Points of inflection are where a curve changes its concavity; from concave up to concave down, or vice versa. These are found by using the second derivative of a function.
To find them, proceed with these steps:

• Compute the second derivative of the function.
• Set the second derivative to zero and solve for the variable.
• Check if these x-values truly correspond to a change in concavity by testing intervals around them.

These points are essential to understand how the graph bends and curves, impacting how a function is interpreted in real-world scenarios.

Derivatives

Derivatives are fundamental in calculus, representing the rate of change of a function. They provide critical information about the behavior of the function at any point.
Here's what you need to know:

• The first derivative indicates the slope of the tangent line at any given point of a function.
• If the first derivative is zero at a point, it’s a possible indicator of a relative extremum.
• The second derivative informs about the concavity of the graph. Positive values mean it’s concave up, and negative values indicate concave down.

Understanding derivatives, through the process of differentiation, forms the basis of analyzing and interpreting different kinds of functions, making calculus an indispensable tool in science and engineering.