Question

In Exercises \(13-20\) , find all points of inflection of the function. $$y=\frac{x^{3}-2 x^{2}+x-1}{x-2}$$

Short answer

The function \(y=\frac{x^{3}-2 x^{2}+x-1}{x-2}\) has no points of inflection because its concavity doesn't change as the second derivative is zero.

Step-by-step solution

Compute the First Derivative

In order to find the first derivative of the function \(y=\frac{x^{3}-2 x^{2}+x-1}{x-2}\), apply the quotient rule that states: \((\frac{u}{v})'\) = \(\frac{vu' - uv'}{v^2}\), where \(u = x^{3}-2 x^{2}+x-1\) and \(v = x-2\). So \(u' = 3x^{2} - 4x + 1\) and \(v' = 1\), and the first derivative becomes \ \(y' = \frac{(x-2)(3x^{2}-4x+1)-(x^{3}-2 x^{2}+x-1)}{(x-2)^2}\)

Simplify the First Derivative

Simplify the obtained expression to ease the following computations. This yields the first derivative \(y' = 2\)

Compute the Second Derivative

The second derivative is derived from the first derivative, and in this case \(y'' = 0\)

Solve the Second Derivative Equals Zero

As the second derivative \(y'' = 0\), implies that x can be any real number inclusive of 2

Check the Changes in the Concavity

As the concavity of a function doesn't change since the second derivative is zero, there are no points of inflection.

Key concepts

First Derivative

Understanding the first derivative of a function is crucial when exploring the dynamics of a function's graph. Essentially, the first derivative, denoted as \(y'\) or \(f'(x)\), reflects the slope or rate of change of the function at any given point along its curve. In practical terms, when you're driving a car, your speedometer is telling you the instantaneous speed of your car at any given moment—that's an example of a derivative in real life!

In the context of the exercise, the first derivative of the function helps us identify where the function is increasing or decreasing, as well as to locate potential relative maxima or minima. It's the stepping stone to finding the concavity of a function and eventually, points of inflection.

Second Derivative

If the first derivative represents the velocity of a function, then the second derivative, often denoted as \(y''\) or \(f''(x)\), can be thought of as the function's acceleration. It tells us about the rate at which the function's slope changes. The second derivative is a powerful tool because it provides insights into the concavity of a graph—whether the graph is curving up or down—and it identifies points of inflection, where the concavity changes.

In our exercise, computing the second derivative was expected to reveal inflection points. However, with \(y'' = 0\), it indicates a constant slope which implies a linear function without concavity, pretty much like a flat road, hence no points of inflection.

Quotient Rule

When you come across a function that is a fraction of two other functions, the quotient rule is your guiding light to finding its derivative. The quotient rule states that the derivative of a function \(\frac{u}{v}\) is given by \(\frac{vu' - uv'}{v^2}\).

Think of the quotient rule as a recipe: to find the derivative of a fraction, you need to take the derivative of the top function (\(u'\)), multiply it by the bottom function (\(v\)), then subtract the product of the top function (\(u\)) and the derivative of the bottom (\(v'\)), and finally, divide everything by the square of the bottom function (\(v^2\)). In this exercise, the quotient rule was the technique employed to compute the first derivative of our more complex rational function.

Concavity of a Function

The concavity of a function gives us a visually graspable idea of how a function curves. It's like choosing between a soup spoon and a tablespoon: the former holds soup better because it's concave up, whereas the latter, convex down, would just spill it. Similarly, a function's graph can be concave up (shaped like a cup) or concave down (shaped like a cap).

A function is concave up where its second derivative is positive, and concave down where it's negative. For the function in our exercise, since the second derivative remains zero for all values of \(x\), the function's concavity does not change, staying consistently linear without forming a cup or a cap shape. Therefore, this informs us that the function lacks points of inflection where the concavity would normally shift.