Problem 29 - Calculus of a Single Variable

Question

Applying the First Derivative Test In Exercises \(17-40\) , (a) find the critical numbers of \(f\) (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. $$ f(x)=(x+2)^{2 / 3} $$

Step-by-Step Solution

Answer

The critical number of the function \(f(x)=(x+2)^{2 / 3}\) is \(x=-2\). The function is decreasing on the interval \((-∞, -2)\) and is increasing on the interval \((-2, ∞)\). The function has a local minimum at \(x=-2\). Check these results using a graphing tool.

Step 1: Find the critical numbers

To find the critical numbers, the first derivatives of the function \(f(x)\) is required, which is found by applying the power rule of differentiation. The derivative of \(f(x)=(x+2)^{2 / 3}\) is \(f'(x)=\frac{2}{3}(x+2)^{-1 / 3}\). Set this equal to zero and solve for \(x\) to find the critical numbers.

Step 2: Determine the intervals of increase and decrease

To determine where the function is increasing or decreasing, consider the sign of the derivative. The function is increasing where \(f'(x)>0\) and decreasing where \(f'(x)<0\).

Step 3: Apply the First Derivative Test

The First Derivative Test states that if \(f'(x)\) changes from positive to negative at a critical number, then \(f(x)\) has a local maximum there. If \(f'(x)\) changes from negative to positive at a critical number, then \(f(x)\) has a local minimum there.

Step 4: Verify with a graphing utility

Use a graphing calculator or an online tool to plot the function and verify the critical numbers, intervals of increase and decrease, and local extrema found in previous steps.

Key Concepts

Critical Numbers

Critical numbers are values of \(x\) where the derivative of a function is either zero or undefined. Finding these points is crucial because they potentially identify where the function could have a local maximum or minimum. To find the critical numbers of our function, \(f(x) = (x + 2)^{2/3}\), we first need to calculate the derivative using the power rule. Differentiating gives us \(f'(x) = \frac{2}{3}(x + 2)^{-1/3}\).

We then solve for \(x\) by setting the derivative equal to zero or considering where it may be undefined. In this example, the derivative \(f'(x)\) is never zero as the expression \(\frac{2}{3}(x + 2)^{-1/3}\) does not reach zero for any real \(x\). However, it is undefined at \(x = -2\) because you cannot divide by zero; hence \(x = -2\) is where the critical number occurs.

Intervals of Increase and Decrease

Intervals of increase and decrease describe where a function's output is getting larger or smaller, respectively. To determine these intervals, we analyze the sign of the derivative \(f'(x)\). The function is increasing where \(f'(x) > 0\) and decreasing where \(f'(x) < 0\).

For our function \(f'(x) = \frac{2}{3}(x + 2)^{-1/3}\), we explore the sign of this expression on either side of the critical point \(x = -2\). For \(x < -2\), \(x + 2\) becomes a negative value raised to a negative power, making the derivative \(f'(x)\) positive, so the function is increasing in this interval.

For \(x > -2\), \(x + 2\) remains positive, causing the derivative \(f'(x)\) to be positive, and thus, the function is still increasing. Therefore, \(f(x)\) is increasing for \(x eq -2\).

Relative Extrema

Relative extrema refer to the local maximum or minimum points within a function. We apply the First Derivative Test to confirm the nature of critical points. This test informs us if a function transitions from increasing to decreasing, or vice versa, at a critical point.

For \(f(x) = (x + 2)^{2/3}\), we have already identified \(x = -2\) as a critical point. Using the First Derivative Test, we observe that there is no sign change in the derivative \(f'(x)\) across \(x = -2\); it is positive on both sides of the critical point. This indicates a saddle point rather than a relative extrema.

Therefore, \(f(x)\) does not have any relative maxima or minima because there is no change from positive to negative or vice versa. This particular finding is crucial to understanding the behavior of the function around critical values.

Graphing Utility Verification

Graphing utilities can visually reassure our algebraic findings. Using a graphing calculator or an online tool is a practical way to verify the findings from the derivative analysis.

By plotting the function \(f(x) = (x + 2)^{2/3}\), we should expect the graph to show increasing behavior everywhere except potentially at the critical number. On the graph, there should be a noticeable change in curvature at \(x = -2\), which aligns with our identification of the saddle point.

This visual check can be enriching: it confirms that \(f(x)\) has no interruptions in its growth, consistent with having a derivative that is positive, except at potentially important points where the nature of the function might change subtly. Seeing these characteristics on the graph enhances understanding and confirms our calculations.