Question

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ y=x^{2}-x-2 $$

Short answer

The graph of the function \(y = x^2 - x - 2\) is concave upward for all x. There is no interval where the function is concave downward.

Step-by-step solution

Find The First Derivative

The first derivative of a function shows the rate at which the function is changing. For the function \(y = x^2 - x - 2\), its derivative \(y' = 2x - 1\) is found by applying the power rule for derivatives to each term.

Find The Second Derivative

The second derivative of a function tells us about the curvature, or 'concavity', of the function. To take the second derivative, we differentiate the first derivative. This gives us \(y'' = 2\).

Determine The Concavity

The second derivative \(y'' = 2\) is positive for all x, since 2 is a positive constant. This means the entire graph for the function is concave upward. The graph is never concave downward because there is no interval where \(y''\) is negative.

Key concepts

First Derivative Test

The first derivative test is an effective method for determining the local maxima and minima of a function. It involves finding the first derivative of the function and identifying where this derivative changes sign. Specifically, this test can be used to determine whether a point is a local maximum, local minimum, or neither.

The steps involve calculating the first derivative, setting it equal to zero to find critical points, and then analyzing the sign of the derivative before and after each critical point. If the sign changes from positive to negative, the critical point is a local maximum; if it changes from negative to positive, it's a local minimum.

In the context of the provided exercise, finding the first derivative, which is \(y' = 2x - 1\), allows us to discover that x = 0.5 is the critical point. This is where the derivative equals zero. However, the first derivative test alone does not provide information about concavity; instead, it's a step toward finding the function's turning points.

Second Derivative Test

The second derivative test offers a reliable way to determine the concavity of a graph and infer points of inflection. Through the second derivative, \(y''\), we ascertain whether a curve is concave upward or concave downward at a particular point.

To apply this test, you first find the second derivative of the function. If \(y'' > 0\) at a specific point, the function is concave upward at that point. Conversely, if \(y'' < 0\), the function is concave downward. If \(y'' = 0\), this may indicate a point of inflection, but further analysis is needed because \(y'' = 0\) can also occur at points that are not inflection points.

In our exercise, the second derivative, \(y'' = 2\), is a positive constant. Thus, the second derivative test concludes that the function \(y = x^2 - x - 2\) is concave upward for all values of x, and no points of inflection exist because there is no change in concavity.

Concave Upward

Concave upward, often visualized as a shape resembling a cup that can hold water, describes a section of a graph where the curve opens upwards. Mathematically, a function's graph is concave upward on intervals where its second derivative is positive. This characteristic indicates that the slope of the tangent line to the curve is increasing.

For instance, the function from our exercise \(y = x^2 - x - 2\) is concave upward at all points because the second derivative \(y'' = 2\) is always positive. Interestingly, concave upward regions of a graph are associated with the function having a local minimum, as seen from the overall shape of the curve.

Concave Downward

Conversely, concave downward refers to a portion of the graph where the curve opens downward like an upside-down cup. A function is concave downward on an interval if its second derivative is negative across that interval. In this case, the slope of the tangent lines is decreasing.

However, in our exercise with the function \(y = x^2 - x - 2\), the second derivative never becomes negative; it is a constant \(y'' = 2\). Therefore, the graph of this function does not exhibit any concave downward behavior. But in general, if you have a function with \(y'' < 0\), you'd expect to find parts of its graph that dip downward, often indicating the presence of a local maximum.