Question

In Exercises \(13-20\) , find all points of inflection of the function. $$y=x^{1 / 3}(x-4)$$

Short answer

The exact solution will depend on the actual solutions to the equation \(x^{-2 / 3} + 2x^{-5 / 3} = 0\). After finding these x-values, confirm they are points of inflection by checking the concavity change and then substitute them back to the original function to find corresponding y-values thus the points of inflection in the form (x, y).

Step-by-step solution

Find the First Derivative

The first derivative of the function \(y = x^{1 / 3}(x-4)\) is found using the product rule. The product rule states that the derivative of the product of two functions is the derivative of the first times the second function plus the derivative of the second function times the first. Thus, \(y' = (1/3)x^{-2/3}(x-4) + x^{1/3}\). Simplifying gives \(y' = (1/3)x^{1 / 3} - (4/3)x^{-2 / 3} + x^{1 / 3}\). Combining like terms gives \(y' = (4/3)x^{1 / 3} - (4/3)x^{-2 / 3}\).

Find the Second Derivative

The second derivative of the function is found by differentiating the first derivative. This gives \(y'' = (4/9)x^{-2 / 3} + (8/9)x^{-5 / 3}\). This can be simplified to \(y'' = (4/9)(x^{-2 / 3} + 2x^{-5 / 3})\).

Find Points of Inflection

Set the second derivative equal to zero and solve for x to find potential points of inflection. This gives \(0 = (4/9)(x^{-2 / 3} + 2x^{-5 / 3})\). Setting the expression inside the brackets equal to zero gives \(x^{-2 / 3} + 2x^{-5 / 3} = 0\). The solutions to this equation are the x-coordinates of potential inflection points.

Check Concavity on Either Side of the Points

To confirm inflection points, check the concavity of the function on either side of the points found in step 3. If the concavity changes sign, then the point is an inflection point. The actual values of the test points are not important, just their signs when plugged into the second derivative.

Calculate Function Values for Inflection Points

After verifying the points of inflection, substitute the x-coordinates of these points into the original function to find respective y-coordinates, thus giving the actual inflection points in the form (x,y).

Key concepts

First Derivative

The first derivative of a function is fundamental in the study of calculus. It represents the rate at which the function's output changes with respect to change in the input. In essence, the first derivative provides information about the function's slope, which helps in determining whether the function is increasing or decreasing at a given point.

For our exercise, we used the product rule to find the first derivative of the function y=x^{1/3}(x-4). The product rule is a technique for differentiating products of two functions, which states that the derivative of a product is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function. Mathematically, if we have two functions u(x) and v(x), the product rule tells us that (uv)' = u'v + uv'.

Applying this rule, the first derivative was calculated as y' = (1/3)x^{-2/3}(x-4) + x^{1/3}, then simplified to get y' = (4/3)x^{1/3} - (4/3)x^{-2/3}. The sign of the first derivative tells us where the function's graph is sloping upward or downward.

Second Derivative

The second derivative of a function gives us information about its concavity, which can indicate the presence of inflection points. When a function's second derivative is positive, the function is concave up, resembling a U-shape. Conversely, when it is negative, the function is concave down, like an upside-down U.

In our exercise, we found the second derivative by differentiating the first derivative. By doing so, we obtained y'' = (4/9)x^{-2/3} + (8/9)x^{-5/3}, which can further be simplified to y'' = (4/9)(x^{-2/3} + 2x^{-5/3}). By setting this derivative equal to zero, we can solve for the x-values that may correspond to inflection points, since these points are where the concavity of the function changes.

Concavity

Concavity refers to the curvature of the graph of a function. If a graph is concave up, it means that it curves upwards, and the function values are above the tangent lines. If a graph is concave down, it curves downward, and the function values are below the tangent lines. The point at which the concavity changes from up to down or down to up is known as an inflection point.

To check for concavity changes, as done in our example, we look at the sign of the second derivative on both sides of the values found by setting the second derivative to zero. A change in sign across these values indicates a change in concavity and confirms the existence of an inflection point. In the exercise, after finding potential inflection points, we verified them by examining the concavity on either side to ensure that there is a concavity change.

Product Rule

The product rule in calculus is a method for finding the derivative of a product of two functions. It states that if we have a function z which is the product of two other functions u and v, such that z = uv, then the derivative of z with respect to x is z' = u'v + uv'. This rule is an essential tool in differentiation as it simplifies the process of finding derivatives of products of functions.

In our problem, the product rule was applied to the function y = x^{1/3}(x-4), to find the first derivative, which is crucial in determining the gradient and potential points of inflection of the function. When applied correctly, using the product rule can greatly simplify the differentiation process and is a cornerstone in the arsenal of calculus techniques.