Question

In Exercises, use a table similar to that in Example 1 to find all relative extrema of the function. $$ f(x)=x^{2}-6 x $$

Short answer

The function \(f(x) = x^{2} - 6x\) has a relative minimum at x = 3.

Step-by-step solution

Find the derivative of the function

Differentiate \(f(x) = x^{2} - 6x\) using the power rule to get \(f'(x) = 2x - 6\). The power rule states that to differentiate \(x^n\), one multiplies the exponent (n) by the coefficient and then reduces the exponent by 1.

Set the derivative equal to zero and solve for x

To find the critical points, where the function could have relative extrema, set \(f'(x) = 2x - 6 = 0\). Solving for x gives \(x = 3\). This is a potential point of relative extrema.

Use a sign table to test the critical point

Choose test points on either side of x = 3 and substitute them into the derivative. If the derivative is positive to the left of 3 and negative to the right, then x = 3 is a local maximum. If it is negative to the left and positive to the right, then x = 3 is a local minimum. For instance, we could choose x = 2 and x = 4. We find that \(f'(2) = -2 < 0\) and \(f'(4) = 2 > 0\), which suggests that f has a local minimum at x = 3.

Key concepts

Derivative of a Function

One of the fundamental concepts in calculus is the derivative of a function. It represents the rate at which the function's output changes with respect to a change in its input. In simpler terms, it's a way to find how steep the slope of the function is at any given point.

The derivative is denoted as f'(x) or df/dx, and it can be calculated using various rules. In the context of our example with the function f(x) = x^2 - 6x, we find the derivative by applying the power rule to each term. Essentially, the derivative gives us a formula to calculate the slope of the curve at any point x.

Critical Points

A critical point occurs where the derivative of a function is either zero or undefined, which can indicate a possible location for relative extrema - that is, local maximums or minimums of the function. In our exercise, after finding the derivative f'(x) = 2x - 6, we determine the critical points by setting the derivative equal to zero and solving the equation. The value x = 3 is a critical point and it's where we focus our search for extrema.

It's important to note that not all critical points will yield a relative extremum. To confirm whether a critical point is indeed an extremum, further analysis, like using a sign chart, is necessary.

Power Rule in Calculus

The power rule is a quick method for differentiating functions in the form x^n, where n is any real number. According to this rule, the derivative of x^n, denoted by f'(x), is nx^(n-1). Thus, when we applied the power rule to each term of our function, we derived f'(x) = 2x - 6 from f(x) = x^2 - 6x.

This rule greatly simplifies the process of finding derivatives for polynomial functions and is one of the first techniques taught to calculus students.

Sign Chart Method

The sign chart method is a tool used to determine the nature of critical points. After identifying potential extrema at the critical points, we test the intervals around these points to understand how the function behaves.

We choose values to the left and right of the critical point and evaluate the derivative with these values. If f'(x) changes sign from positive to negative as we pass through the critical value, the function has a local maximum there. Conversely, if f'(x) changes from negative to positive, then the function has a local minimum. In our example, evaluating f'(x) at points around x = 3 revealed a change from negative to positive, indicating a local minimum at x = 3.