Question

Identify the open intervals on which the function is increasing or decreasing. $$ g(x)=x^{2}-2 x-8 $$

Short answer

The function \(g(x) = x^{2} - 2x - 8\) is decreasing in the interval \((-∞, 1)\) and increasing in the interval \((1, ∞)\).

Step-by-step solution

Find the Derivative

First, find the derivative of the function \(g'(x)\) using the power rule. \(g'(x) = 2x - 2\).

Find the Critical Points

Set the derivative equal to zero, i.e., solve \(g'(x) = 0\), to find the critical points. Therefore \(2x - 2 = 0\) gives \(x = 1\). So, the critical point is \(x = 1\).

Determine the Intervals of Increase and Decrease

Check the sign of the derivative \(g'(x)\) in the intervals \((-∞, 1)\) and \((1, ∞)\). If \(g'(x)\) is positive, the function is increasing and if \(g'(x)\) is negative, the function is decreasing. For \(x < 1\), \(g'(x) < 0\), so the function is decreasing. For \(x > 1\), \(g'(x) > 0\), so the function is increasing.

Key concepts

Derivative of a Function

The derivative of a function is a foundational concept in calculus that measures how a function's output value changes as its input value changes. In practical terms, it gives the slope of the tangent line at any point on the function's graph. This slope can be thought of as the function's 'rate of change' at that specific point.

When evaluating the behavior of functions like the given quadratic function, the derivative plays a crucial role. Calculating the derivative, as in our example with function g(x) = x^2 - 2x - 8, involves applying rules of differentiation to find g'(x), which describes how the function changes at every point along its domain. The process of finding the derivative, as demonstrated in the solution, is fundamental for further exploration of the function's properties, such as increasing and decreasing intervals.

The derivative is not only a tool for analysis; it also has practical applications in various fields like physics, engineering, and economics, where understanding the rate at which things change is crucial.

Critical Points

Critical points are certain points on a function's graph where the derivative is either zero or undefined. These points are 'critical' because they often indicate something notable about the function's behavior, like local maxima, local minima, or points of inflection.

In the context of our example, identifying the critical point(s) is step two in analyzing where function g(x) is increasing or decreasing. By setting the derivative equal to zero and solving for x, we find that x = 1 is a critical point for g(x). What this means is that the slope of the tangent to the curve at x = 1 is zero, which could imply that the function has a flat spot or a peak/valley at that point. To conclude whether it's a maximum or minimum or neither, one might need further investigation using second derivative tests or other methods. However, for the purpose of determining increasing and decreasing intervals, knowing where the derivative is zero gives us the essential boundaries needed to test intervals around these critical points.

Power Rule

The power rule is one of the most basic and commonly used techniques for finding derivatives. To apply the power rule, if you have a function of the form f(x) = x^n, where n is any real number, the derivative of that function is f'(x) = n*x^(n-1).

Let's look at the given function in the exercise, g(x) = x^2 - 2x - 8. When you apply the power rule to the term x^2, the derivative becomes 2x. The term -2x is already linear, and its derivative by applying the power rule is -2. Constants like -8 have a derivative of zero, since they do not change as x changes. Therefore, using the power rule, the derivative g'(x) results in 2x - 2.

The power rule simplifies the process of finding gradients of polynomial functions, making them far more approachable when determining the behavior of the function, such as finding increasing and decreasing intervals.