Problem 1 - Essential Calculus

Question

In Exercises \(1-8\), determine the open intervals on which the graph is concave upward or concave downward. \(y=x^{2}-x-2\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = x^{2}-x-2\) is concave upward for all \(x\).

Step 1: Find the First Derivative

The first derivative of the function \(f(x) = x^{2}-x-2\) can be found using the power rule which states that the derivative of \(x^n\) is \(nx^{n-1}\). Therefore, the first derivative \(f'(x) = 2x - 1\).

Step 2: Find the Second Derivative

The second derivative is the derivative of the first derivative. Applying the power rule again, we get \(f''(x) = 2\).

Step 3: Analyze the Second Derivative

Since the second derivative \(f''(x) = 2\) is positive for all \(x\), this implies that the function is concave upwards for all values of \(x\). No inflection points exist because the sign of the second derivative never changes.

Key Concepts

Second Derivative

The second derivative of a function can reveal a wealth of information about the function's graph, particularly with respect to its concavity. It is obtained by differentiating the first derivative of the function. For the parabola described by the function f(x) = x^2 - x - 2, we first find the first derivative f'(x) = 2x - 1 using the power rule of differentiation.

From here, we apply the power rule once more to find the second derivative, denoted as f''(x), resulting in a constant f''(x) = 2. A positive second derivative indicates that the original function is concave upward everywhere. Conversely, a negative second derivative would indicate that the function is concave downward. Since the second derivative in our example is a constant positive value, we can conclude that the function's graph is a concave upward parabola at all points.

Concave Upward

When the graph of a function is described as 'concave upward,' imagine a shape like a smile or the interior of a bowl facing upwards. Mathematically, a function is concave upward on an interval if its second derivative is positive within that interval.

In the case of f(x) = x^2 - x - 2, since the second derivative f''(x) = 2 is positive for all values of x, the graph of this function forms a 'smile,' indicating that it is concave upward on the entire real number line. This shape suggests that the rate of change of the slope (the first derivative) is increasing, much like how the slope of the hill increases as you move uphill.

Power Rule

Understanding the power rule is essential for differentiating polynomials. The power rule states that for a given function of the form f(x) = x^n, where n is a real number, its derivative with respect to x is f'(x) = nx^{n-1}.

In our function f(x) = x^2 - x - 2, we apply the power rule to each term that contains an x raised to a power. The term x^2 yields a derivative of 2x, and -x becomes -1. Constants such as -2 have a derivative of zero because they do not change as x changes. The power rule simplifies the process of finding the derivatives we need to analyze the graph's concavity.

Inflection Points

Inflection points are fascinating features on a graph where the concavity changes from upward to downward or vice versa. They represent points where the second derivative of the function changes sign. For example, if the second derivative goes from positive to negative, the graph switches from concave upward to concave downward.

In the context of the function f(x) = x^2 - x - 2, our analysis showed that the second derivative f''(x) = 2 is always positive; it does not change signs. This means there are no inflection points since the graph does not switch concavity. Had there been a variable x in the second derivative, we might have found values of x where the sign changes, thus indicating potential inflection points. Inflection points are crucial in understanding the overall shape and turning points of a graph.