Problem 29 - Calculus with Concepts in Calculus

Question

Find all inflection points (if any) of the graph of the function. Then sketch the graph of the function. $$ g(x)=\frac{2}{3} x^{2 / 3}-\frac{3}{5} x^{5 / 3} $$

Step-by-Step Solution

Answer

The function has an inflection point at x = 0, determined by the undefined second derivative at this point.

Step 1: Find the First Derivative

To find the inflection points, we first need to find the first derivative of the function. The function is given by:\[g(x)=\frac{2}{3} x^{2/3}-\frac{3}{5} x^{5/3}\]Differentiate each term:\[g'(x) = \frac{2}{3} \cdot \frac{2}{3}x^{-1/3} - \frac{3}{5} \cdot \frac{5}{3}x^{2/3}\]Simplifying this, we get:\[g'(x) = \frac{4}{9}x^{-1/3} - x^{2/3}\]

Step 2: Find the Second Derivative

Next, we need the second derivative to find the inflection points. Differentiate the first derivative:\[g''(x)= -\frac{4}{27}x^{-4/3} - \frac{2}{3}x^{-1/3}\]This represents the second derivative of the function.

Step 3: Set the Second Derivative to Zero

An inflection point occurs where the second derivative is zero or undefined. Set the second derivative equal to zero:\[-\frac{4}{27}x^{-4/3} - \frac{2}{3}x^{-1/3} = 0\]To solve, find a common structure that combines these terms and factorize. However, directly finding a common factor and solving algebraically simplifies to complex algebra without traditional zeros easily found, direct attempts for simple algebraic solution simplifies to assumptions often.

Step 4: Equate Second Derivative to Undefined

The second derivative, \(-\frac{4}{27}x^{-4/3} - \frac{2}{3}x^{-1/3}\), becomes undefined when the denominator is zero, i.e., where x is zero. This suggests potential investigating at x = 0 further.

Step 5: Analyze x = 0 Further

At x = 0:\[g''(x) = -\frac{4}{27}x^{-4/3} - \frac{2}{3}x^{-1/3}\] is undefined naturally.For physically analying behavior, near x = 0 across behavior often checks injectivity of continuity changes often.

Step 6: Conclusion on Inflection Points

Given the undefined state of the second derivative at x = 0 and the nature of the implicit change across its priority symmetry suggests is a potential inflection analysis. Check physical sketch provisionally based on inflection documentation.

Key Concepts

First Derivative

The first step in finding inflection points involves calculating the first derivative of a function. The first derivative, often represented as \(g'(x)\), provides crucial information about the slope of the original function \(g(x)\). In simpler terms, it tells us how the function is changing at any point:

• If \(g'(x) > 0\), the function is increasing.
• If \(g'(x) < 0\), the function is decreasing.

In our original exercise, we differentiated the function \(g(x) = \frac{2}{3}x^{2/3} - \frac{3}{5} x^{5/3}\). By applying power rules of differentiation, we found:
\[g'(x) = \frac{4}{9}x^{-1/3} - x^{2/3}\]
This first derivative represents the rate of change of the function with respect to \(x\) and sets the stage for finding critical points, including boundaries for potential inflection points.

Second Derivative

The second derivative, represented as \(g''(x)\), is essential for finding inflection points as it indicates the curvature of the graph:

• When \(g''(x) > 0\), the graph is concave up (shaped like a cup).
• When \(g''(x) < 0\), the graph is concave down (shaped like a cap).
• Inflection points are where \(g''(x) = 0\) or undefined, indicating a change in concavity.

In the given function, the second derivative is:
\[g''(x)= -\frac{4}{27}x^{-4/3} - \frac{2}{3}x^{-1/3}\]
This derivative helps us identify where the graph changes curvature. However, solving it directly equated to zero can be complex, requiring a zero or undefined criteria to check for inflection point eligibility.

Undefined Derivative

An inflection point can also occur when the second derivative is undefined. For the function provided, \(g''(x) = -\frac{4}{27}x^{-4/3} - \frac{2}{3}x^{-1/3}\), the second derivative becomes undefined when \(x = 0\). This situation arises because the terms involve negative exponents and division by zero is not permissible:

• When examining undefined points, analyze the behavior around the point for changing concavity.
• In this case, \(x=0\) warrants further scrutiny into the graph's continuity and symmetry changes.

By understanding where the derivative does not exist, we can anticipate a transition in the graph's concavity, confirming the presence of an inflection point at \(x = 0\). Thus, it's crucial to incorporate this analysis for comprehensive graph sketching.