Problem 27 - Calculus A New Horizon

Question

Give a complete graph of the function, and identify the location of all critical points and inflection poir s. Check your work with a graphing utility. $$2 x+3 x^{2 / 3}$$

Step-by-Step Solution

Answer

The critical points are at \(x = -1\) and \(x = 0\), with a potential inflection point at \(x = 0\). Use a graphing utility to confirm.

Step 1: Define the function

The given function is \( f(x) = 2x + 3x^{2/3} \). We'll analyze this function to find critical points and inflection points.

Step 2: Find the derivative

To identify critical points, we first need to find the derivative of \( f(x) \). The derivative is \( f'(x) = \frac{d}{dx}(2x) + \frac{d}{dx}(3x^{2/3}) = 2 + 2x^{-1/3} \).

Step 3: Set the derivative to zero and solve for x

Critical points occur where the derivative is zero. Set \( f'(x) = 0 \): \[ 2 + 2x^{-1/3} = 0 \]Simplify to find:\[ 2 = -2x^{-1/3} \] \[ x^{-1/3} = -1 \]\[ x = (-1)^{-3} = -1 \] Thus, \( x = -1 \) is a critical point.

Step 4: Check the derivative for undefined points

Since \( f'(x) = 2 + 2x^{-1/3} \), it is undefined when \( x = 0 \) because the term \( x^{-1/3} \) is not defined at zero. This makes \( x = 0 \) another critical point.

Step 5: Find the second derivative

To identify inflection points, find the second derivative: \[ f''(x) = \frac{d}{dx}(2 + 2x^{-1/3}) = -\frac{2}{3}x^{-4/3} \].

Step 6: Set the second derivative to zero and solve for x

Inflection points occur where the second derivative is zero or undefined. However, \( -\frac{2}{3}x^{-4/3} = 0 \) has no real solutions. The second derivative is undefined at \( x = 0 \), so this could be an inflection point.

Step 7: Analyze the function with a graphing utility

Graph the function \( f(x) = 2x + 3x^{2/3} \) using a graphing utility to confirm the critical and inflection points. The graph should confirm the points \((x = -1, x = 0)\) as critical points and potentially \(x = 0\) as an inflection point.

Key Concepts

Inflection Points

Inflection points are locations on a graph where the curvature changes direction. In mathematical terms, this is where the second derivative of a function changes sign. During the analysis of the function \( f(x) = 2x + 3x^{2/3} \), we determined that the second derivative \( f''(x) = -\frac{2}{3}x^{-4/3} \) becomes undefined at \( x = 0 \).
However, when attempting to set \( f''(x) \) equal to zero and solving for \( x \), there were no real solutions. Therefore, the only potential inflection point based on the second derivative is where it is undefined. This critical analysis suggests that \( x = 0 \) may indeed be an inflection point.
To confirm this further, reviewing the graph of the function can provide insight. A graphing utility would illustrate a change in the concavity of this curve around \( x = 0 \), supporting the potential existence of an inflection point.

Graphing Functions

Graphing functions offers a visual representation of the behavior of a mathematical function which can show where critical and inflection points occur. Using a graphing utility allows you to input the function \( f(x) = 2x + 3x^{2/3} \) and observe its shape and key features.
When you graph this function, you can visually identify the critical points and check potential inflection points. Observing that the function appears to say a curve change around \( x=0 \) would visually verify the previously calculated results.

• At \( x = -1 \), the graph will show a slope of zero, representing a critical point.
• Another critical point at \( x = 0 \) can also be noted where the derivative does not exist.
• The visual cue of changing curvature near \( x = 0 \) helps to confirm it as an inflection point.

Graphing is a powerful tool, complementing the analytical findings by providing a straightforward method to double-check the mathematical calculations.

Derivative Analysis

Derivative analysis involves seeking out critical points of a function, which occur where the first derivative equals zero or is undefined. The function \( f(x) = 2x + 3x^{2/3} \) shows this real-life application of derivative analysis.
First, the derivative \( f'(x) = 2 + 2x^{-1/3} \) is determined. To find critical points, set the derivative to zero: \[ 2 + 2x^{-1/3} = 0 \]Solving this yields \( x = -1 \). This point is significant as it is where the slope of the tangential line to the curve is zero.
Additionally, derivative analysis highlights where the derivative is undefined. For this function, \( f'(x) \) is undefined at \( x=0 \), marking it as a critical point too. Derivative analysis thus helps identify noteworthy points that graphing or further study will shed light on, such as potential inflection points.