Problem 27 - University Calculus: Early Transcendentals

Question

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$h(x)=\sqrt[3]{x}, \quad-1 \leq x \leq 8$$

Step-by-Step Solution

Answer

Minimum: -1 at x = -1; Maximum: 2 at x = 8.

Step 1. Understand the Function and Interval

The function given is \( h(x) = \sqrt[3]{x} \). We are tasked to find its absolute maxima and minima within the interval \(-1 \leq x \leq 8\). The cube root function is increasing over all real numbers.

Step 2. Evaluate the Function at Endpoints

Calculate the function values at the interval endpoints: \(-1\) and \(8\).- \(h(-1) = \sqrt[3]{-1} = -1\) - \(h(8) = \sqrt[3]{8} = 2\)

Step 3. Check for Critical Points

Find critical points by taking the derivative of \(h(x)\) and setting it to zero or undefined. Since \(h(x) = x^{1/3}\), derivative \(h'(x) = \frac{1}{3}x^{-2/3}\). The derivative is never zero, and it's undefined at \(x=0\) which doesn't appear as a critical point since \(0\) is not in \((-1,8)\).

Step 4. Compare Function Values

The potential locations for absolute extrema are the endpoints and any critical points. Since there's no critical point in the interval, only the endpoints need evaluation. Compare:- \(h(-1) = -1\)- \(h(8) = 2\)Thus, absolute minimum is \(h(x) = -1\) at \(x = -1\) and absolute maximum is \(h(x) = 2\) at \(x = 8\).

Step 5. Sketch the Graph

Graph \( h(x) = \sqrt[3]{x} \) over the interval \(-1 \leq x \leq 8\). Mark the endpoints:- Point \((-1, -1)\) where the absolute minimum occurs. - Point \((8, 2)\) where the absolute maximum occurs.

Key Concepts

Absolute Extrema

In calculus, the concept of absolute extrema helps us find the highest and lowest points on a graph of a function over a specified interval. These points are crucial as they tell us the overall maximum or minimum value that the function takes within that interval.
To determine these points, we need to:

• Assess the function values at the endpoints of the interval.
• Evaluate any critical points within the interval, though in this exercise, the cube root function has no critical points due to the nature of its derivative.

For the function given, \( h(x) = \sqrt[3]{x} \), under the interval \(-1 \leq x \leq 8\), the work primarily involves checking the values at \(x = -1\) and \(x = 8\).
The absolute minimum value is found at \(x = -1\) and it equals \(-1\). Similarly, the absolute maximum value occurs at \(x = 8\), equaling \(2\). These values confirm the extrema since no other points, specifically critical points, exist within the interval due to the nature of the derivative.

Derivative

The derivative is a fundamental concept in calculus that reveals the rate of change of a function with respect to a variable. For the function \( h(x) = \sqrt[3]{x} \), we derive the expression to explore any critical points within the interval.
The derivative, noted as \( h'(x) \), is calculated using the power rule: \[ h'(x) = \frac{1}{3}x^{-2/3} \]This derivative describes the slope of the function at any given point, indicating the function is always increasing except where it's undefined at \(x = 0\).

• A critical point is where the derivative is zero or doesn't exist in the given interval.
• In our case, the derivative didn't yield critical points to evaluate because the derivative doesn't equal zero anywhere within \(-1 \leq x \leq 8\).

Since the derivative is undefined at \(x = 0\), which lies within the interval, it might initially seem significant. However, since it's not included in the interval endpoints, it doesn't count as a critical point for this problem.

Function Graph

Graphing a function is a visual way to understand how it behaves over a given interval. By plotting \( h(x) = \sqrt[3]{x} \) within the interval \(-1 \leq x \leq 8\), we see how it changes.
Here are some steps to graph this function:

• Plot the endpoints: \((-1, -1)\) for the absolute minimum and \((8, 2)\) for the absolute maximum.
• The cube root function naturally curves upwards, indicating an increasing nature throughout the interval.

The graph highlights how the function smoothly ascends from left to right. This clear upward trend on the graph represents that \( h(x) \) steadily grows without interruption or internal changes in direction in this interval.
When identifying these key points \((-1, -1)\) and \((8, 2)\), you help confirm both the absolute minimum and maximum observed in the function's values at the interval's endpoints. Graphing gives us not just solutions to a problem but an intuitive grasp of how and why these points matter.