Problem 17 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=x^{2}-6 x $$

Step-by-Step Solution

Answer

The critical number of \(f(x) = x^{2} - 6x\) is \(x = 3\). The function is decreasing for \(x < 3\) and increasing for \(x > 3\). There is a relative minimum at \(x = 3\).

Step 1: Find the derivative of \(f(x)\)

The derivative of \(f(x) = x^{2} - 6x\) is \(f'(x) = 2x - 6\).

Step 2: Find the critical numbers

Setting the derivative equal to zero gives \(2x - 6 = 0\). Solving this for \(x\) gives \(x = 3\), so the function only has one critical number.

Step 3: Determine where \(f(x)\) is increasing or decreasing

Choose test points in the intervals \(x < 3\) and \(x > 3\). For \(x < 3\), choose \(x = 2\). Substituting into the derivative gives \(f'(2) = -2\), so \(f(x)\) is decreasing on this interval. For \(x > 3\), choose \(x = 4\). Substituting into the derivative gives \(f'(4) = 2\), so \(f(x)\) is increasing on this interval.

Step 4: Find the relative extrema

The function has a relative minimum at \(x = 3\) because the function changes from decreasing to increasing at this point.

Step 5: Confirm the results using a graphing utility

Graph \(f(x) = x^{2} - 6x\). The graph should have a minimum point at \(x = 3\), confirming the previous results.

Key Concepts

Derivative of a Function

The derivative of a function represents the rate at which the function's values change with respect to changes in its input. In simpler terms, it can be thought of as the function's slope at any given point. For the function f(x) = x^2 - 6x, its derivative f'(x) = 2x - 6 tells us how f(x) is changing at each value of x. Concretely, if f'(x) > 0, the function is increasing; if f'(x) < 0, it is decreasing; and if f'(x) = 0, we may have found a critical point, such as a peak or a trough.
To find this derivative, we used basic rules of differentiation, such as the power rule. These rules give us the tools to analyze the behavior of functions and predict their graphs without a calculator.

Increasing and Decreasing Intervals

When we talk about increasing or decreasing intervals of a function, we are referring to sections of the function’s domain where its output either consistently goes up or down. A function is considered to be increasing on an interval if the derivative is positive throughout that interval, and decreasing if the derivative is negative. In the case of the function f(x) = x^2 - 6x, by testing values to the left and right of our critical point, we found that the function is decreasing when x < 3 and increasing when x > 3. Knowing where a function increases and decreases is crucial for understanding its overall shape and behavior.

Choosing Test Points

Choosing an appropriate test point within an interval can quickly tell us the behavior of the function without having to analyse the whole interval. It’s a practical step for gaining insight into a function’s behavior efficiently.

Relative Extrema

Relative extrema refer to the high and low points on a graph within a particular region. A relative minimum is the lowest point in its neighborhood, and a relative maximum is the highest. By analyzing where a function changes from decreasing to increasing, we can locate a relative minimum, and vice versa for a maximum. In our example with f(x) = x^2 - 6x, the function switches from decreasing to increasing at x = 3. This tells us that there’s a relative minimum at that point.
A powerful aspect of calculus is identifying these points by merely using derivatives, which can reveal a lot about a function’s graph without the need to plot numerous points manually.

Graphing Utility in Calculus

Graphing utilities are invaluable tools in calculus for visualizing functions and their derivatives. They can confirm analytical results such as critical points, and relative extrema, and visually depict increasing and decreasing intervals. Once we've used calculus techniques to predict the behavior of f(x) = x^2 - 6x, a graphing utility helps to visually validate our findings. It displays the decreasing interval as the slope heads downwards to the left of x = 3 and the increasing interval as the slope goes upwards to the right. The relative minimum at x = 3 is also clearly visible as the lowest point in the graph around that area. This step serves as a powerful check against our computations and offers a graphical understanding that can reinforce the concept.