Question

Identify the open intervals on which the function is increasing or decreasing. $$ y=x+\frac{4}{x} $$

Short answer

The given function \(y = x + \frac{4}{x}\) is increasing on the intervals \((-2, 0)\) and \((2, \infty)\), and decreasing on the intervals \((- \infty, -2)\) and \( (0,2) \).

Step-by-step solution

Calculating the Derivative

To find where the function increases or decreases, we need to find its derivative. The derivative of \(y = x + \frac{4}{x}\) can be calculated as follows: Using the power rule of derivatives \(\frac{d}{dx}x^n = nx^{n-1}\) and the fact that \( \frac{4}{x} = 4x^{-1}\), we get: \(y' = 1 - 4x^{-2} = 1 - \frac{4}{x^2}\).

Finding the Critical Points

The critical points of a function are where its derivative equals zero or is undefined. To find the critical points for this function, set the derivative equal to zero and solve for x: \(1 - \frac{4}{x^2} = 0 \rightarrow \frac{4}{x^2}=1 \rightarrow x^2 = 4 \rightarrow x = \pm 2\), thus, the critical points are \(x = 2\) and \(x = -2\). Note that the derivative is defined for all \(x \neq 0\).

Using the Number Line to Find Intervals of Increase and Decrease

Now that we know the critical values, we can construct a number line with these values and choose test points in each interval to check whether the function increases or decreases. Choose test points \(x = -3, -1, 0, 1, 3\). Calculating the derivative at these points results in \(y'(-3) = -5\), \(y'(-1) = 5\), \(y'(0)\) is undefined, \(y'(1) = -3\), and \(y'(3) = 5/3\). Therefore, the function is decreasing (y'<0) on the interval \((- \infty, -2)\), increasing (y'>0) on intervals \((-2, 0)\) and \((2, \infty)\), and decreasing on the interval \((0,2)\).

Key concepts

Derivative

The derivative of a function can be thought of as the equation for the slope of a tangent line at any point on the function. It represents the rate at which the function's output is changing at any given point relative to its input. In simpler terms, the derivative helps us understand how fast or slow a function is increasing or decreasing at various points.

For example, if you're driving and your speedometer reads 60 miles per hour, you can think of this as the derivative of your position with respect to time. It tells you how quickly your location is changing.

Mathematically, derivatives are found by taking the limit of the difference quotient as the difference in the input values approaches zero, which is often taught in calculus using rules like the power rule and the product/quotient rules for differentiation.

Critical Points

Critical points are essential in the study of calculus because they help identify where a function's graph has potential maximums, minimums, or inflection points. A critical point occurs where the derivative of a function is either zero or undefined.

Why is this important? Imagine you're on a roller coaster; the highest and lowest points where the ride changes direction would be considered critical points of your journey. In the context of a graph, these points mark where the function switches from increasing to decreasing or vice versa.

Identifying these points involves setting the derivative equal to zero or finding where the derivative does not exist. Analyzing what happens at and around these points can tell us a lot about the function's behavior.

Power Rule

The power rule is a fundamental technique in calculus for finding the derivative of a function in the form of a power. The rule states that if you have a function like \(x^n\), where \(x\) is the base and \(n\) is the exponent, then the derivative of that function is \(nx^{n-1}\).

Think of it as a shortcut; instead of using the definition of the derivative which involves limits, the power rule lets you quickly find the derivative of polynomials and any function that can be represented as a power of \(x\). This means, for a function \(y=x^n\), if you want to know the ‘speed’ at which \(y\) changes with respect to \(x\), you simply multiply the exponent by the base and decrease the exponent by one.

It’s like having a tool that instantly tells you the steepness of a hill at any point, just by knowing its shape. This is especially helpful when dealing with more complex functions.

Test Points

Test points are specific values that you choose from within the intervals created by the critical points. They serve as samples to determine the behavior of a function—whether it's increasing or decreasing—in various segments.

Picture a treasure map with 'X' marks denoting critical spots. Test points would be steps you take towards those 'X' marks to make sure you're going in the right direction. They’re like dipping your toes in water to check the temperature before jumping in. By evaluating the derivative at these test points, you can conclude whether the function is climbing uphill or rolling downhill in that interval.

This approach provides a practical way to visualize the function's behavior without plotting the entire graph. Through the use of test points, one can efficiently sketch the increase or decrease trend of a function over its domain.