Problem 18 - Calculus of a Single Variable

Question

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=(x+1)^{2}-3(x+1)^{2 / 3} $$

Step-by-Step Solution

Answer

The analysis and sketching of the function involve finding its intercepts, extrema, inflection points and checking for asymptotes. The intercepts are computed by equating y to zero, extrema are found utilizing first derivative and second derivative for inflection points. Verification of results will be done with a graphical representation.

Step 1: Finding the Intercepts

To find the intercepts, first set \(y = 0\) and solve for \(x\). So, \(0 = (x+1)^{2} - 3(x+1)^{2/3}\). The roots from this equation represents the x-intercepts of the function.

Step 2: Finding the Extrema

By differentiation, the relative extrema can be found. Extrema occur where the derivative either equals zero or is undefined. The first derivative of the function is: \(y' = 2(x+1) - 2(x+1)^{-1/3}\). Set \(y' = 0\), and find the solutions for \(x\), these represent locations of potential relative extrema.

Step 3: Finding Points of Inflection

Inflection points can be found by utilizing the second derivative test. These are the points where concavity changes. The second derivative of the function is: \(y'' = 2 + 2/3(x+1)^{-4/3}\). Set \(y'' = 0\) and find the solutions for \(x\), these points are potential points of inflection.

Step 4: Verification with Graphing Utility

By using a graphing utility, verify all the calculations done in previous steps. This will allow visual confirmation of the intercepts, relative extrema and points of inflection previously obtained. Check if there are any asymptotes present in the graph.

Key Concepts

Intercepts

Intercepts are the points where the graph crosses the axes. To find the x-intercepts, we set the function equal to zero, and solve for x. In other words, we solve the equation \(0 = (x+1)^{2} - 3(x+1)^{2/3}\). These x-values, where the function equals zero, are where the graph touches or crosses the x-axis. If we also want the y-intercept, which is the point where the graph crosses the y-axis, we substitute \(x = 0\) into the function and solve for y. This will give us a point in the form of (0, y).

A quick recap of the steps:

• For x-intercepts, solve \((x+1)^{2} - 3(x+1)^{2/3} = 0\).
• For the y-intercept, calculate \(y\) when \(x = 0\).

Finding the intercepts is the first step in understanding where your graph will sit relative to the axes.

Relative Extrema

Relative extrema are points where the function reaches a local maximum or minimum, meaning the graph changes direction at these points. To find them, you'll need to calculate the first derivative of the function. Once you have the derivative, you'll set it equal to zero or look for where it is undefined. Solving these conditions will give you the x-values of potential extrema.

For our function:

• The first derivative is \(y' = 2(x+1) - 2(x+1)^{-1/3}\).
• Set \(y' = 0\) and solve for x to find potential points of relative extrema.

These x-values indicate where the slope of the tangent to the graph is zero, signifying a peak or trough in the graph. Be careful to verify these points can indeed be relative extrema by ensuring the second derivative test or another method confirms it.

Points of Inflection

Points of inflection are critical in understanding the nature of a graph as they are where the concavity changes. A change in concavity means the graph switches from being 'smiley' to 'frowny' or vice versa. To find these points, calculate the second derivative of the function, and solve \(y'' = 0\).

For the given function, the second derivative is

• \(y'' = 2 + \frac{2}{3}(x+1)^{-4/3}\).
• Set \(y'' = 0\) and solve for \(x\) to find the x-values of potential inflection points.

These are the points where the rate of change of the slope itself changes, hinting at a twist or change in the curve's direction. It's important to have a graphing utility handy to visualize these changes for confirmation.

Asymptotes

Asymptotes are lines that a graph approaches but never truly meets. They are more common in certain types of functions, such as rational functions, but it's useful to check for both vertical and horizontal asymptotes in most graphs.

While our specific function does not inherently result in vertical asymptotes due to its polynomial nature, it's still a good practice to inspect potential horizontal asymptotes. Using limits as \(x\) approaches infinity or negative infinity, we can discern whether the function flattens out in any direction.

Here’s how you break it down:

• Check for vertical asymptotes by finding values of \(x\) that make the function undefined.
• Inspect horizontal asymptotes by evaluating the limits of the function as \(x\) goes to infinity or negative infinity.

This concept helps us complete the picture of the function's end behavior and how the function extends across the axes in a larger scope.