Problem 30 - Thomas Calculus

Question

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. \begin{equation} y=x^{2 / 5} \end{equation}

Step-by-Step Solution

Answer

The absolute minimum is at \((0, 0)\), and there are no inflection points or local extrema other than the minimum at the origin.

Step 1: Find the First Derivative

The first step to find the extreme points and inflection points is to determine the first derivative of the function. For the given function \(y = x^{2/5}\), the derivative \(y'\) is calculated by using the power rule. The derivative is given by \( \frac{d}{dx}(x^{n}) = nx^{n-1} \). Therefore, \(y' = \frac{2}{5}x^{-3/5}\).

Step 2: Set the First Derivative to Zero

To find the critical points, set the first derivative \(y' = \frac{2}{5}x^{-3/5}\) equal to zero and solve for \(x\). However, this expression equals zero when the numerator is zero, and since \(\frac{2}{5}\) is never zero, there are no solutions for which \(y'\) equals zero.

Step 3: Analyze the First Derivative for Local Extrema

Since \(y' = \frac{2}{5}x^{-3/5}\) is never zero, there are no local extrema where the slope of the tangent (the first derivative) is zero. However, we need to consider the endpoints and the behavior of undeclared points such as \(x = 0\) due to the power of \(x\).

Step 4: Check the Domain and Endpoints for Absolute Extrema

The given function \(y = x^{2/5}\) is defined for all non-negative \(x\) (since negative bases for real-power roots like \(2/5\) involve complex numbers). Evaluate the function at \(x = 0\), giving \(y(0) = 0^{2/5} = 0\). As \(x\) approaches infinity, \(y = x^{2/5}\) increases without bound.

Step 5: Find the Second Derivative

To identify inflection points, calculate the second derivative \(y''\). Differentiate the first derivative \(y' = \frac{2}{5}x^{-3/5}\) again to find \(y'' = \frac{d}{dx}(\frac{2}{5}x^{-3/5}) = \frac{-6}{25}x^{-8/5}\).

Step 6: Analyze the Second Derivative for Inflection Points

Set \(y'' = \frac{-6}{25}x^{-8/5} = 0\) to find potential inflection points. Since the expression involves a constant in the numerator and \(x^{-8/5}\) in the denominator which is never zero (as long as \(xeq0\)), there are no inflection points in the domain of the function. However, the behavior around \(x = 0\) is important. \(y''\) is undefined at \(x = 0\), indicating a possible change in concavity there.

Step 7: Sketch the Graph

To understand the overall behavior, plot the function. As \(x\rightarrow 0^{+}\), \(y = x^{2/5}\) flattens to zero, and the curve becomes steeper as \(x\) increases. The function is continuously increasing for all positive \(x\), with \(x = 0\) as an absolute minimum point.

Key Concepts

Derivative Calculation

The foundation of understanding the behavior of a function often involves finding its derivatives. The derivative tells us about the rate at which the function's value changes as the input changes.
For the function \(y = x^{2/5}\), we use the power rule to calculate the first derivative. This rule states that \(\frac{d}{dx}(x^n) = nx^{n-1}\) for any real number \(n\). Applying this to \(y = x^{2/5}\), we get:
\[ y' = \frac{2}{5}x^{-3/5} \]
The negative exponent indicates that \(y'\) involves a division by \(x^{3/5}\), highlighting how the derivative behaves for different \(x\) values.

Critical Points

Critical points are values of \(x\) where the derivative either equals zero or is undefined. These points can represent local maxima, minima, or saddle points.
To find critical points for the function \(y = x^{2/5}\), we set the first derivative \(y' = \frac{2}{5}x^{-3/5}\) to zero:
- Since the numerator is a constant \(\frac{2}{5}\), \(y' = 0\) has no solution as the numerator cannot be zero.
- We must, however, consider where \(y'\) is undefined. Since \(y'\) is undefined at \(x = 0\), \(x = 0\) becomes a critical point. The function attains an absolute minimum at this point, as the value of \(y\) is zero (\(y(0) = 0^{2/5} = 0\)).

Domain of Function

Understanding the domain of a function helps identify where it is defined and where certain calculations, like derivatives, apply.
For \(y = x^{2/5}\), the domain includes all non-negative values of \(x\) (\(x \geq 0\)).
- Negative values of \(x\) are not in the domain since the function involves a real-number power root, which isn't defined for negative bases in real number calculations.
- This domain consideration ensures that derivative computations are only done for values where the original function is valid and avoids any conceptual errors when analyzing the function.

Second Derivative Test

The second derivative test helps in determining the concavity of a function and locating inflection points, where the function changes concavity.
For the function \(y = x^{2/5}\), the second derivative \(y''\) is calculated from the first derivative. Differentiating again yields:
\[ y'' = \frac{-6}{25}x^{-8/5} \]
To find inflection points, set \(y'' = 0\). However, \(y''\) has a constant numerator and is never zero. Therefore, there are no straightforward inflection points.
- Importantly, \(y''\) is undefined at \(x = 0\). This indicates a potential change in concavity at \(x = 0\), suggesting the function's curve may change shape there. Understanding these nuances aids in graphing and analyzing the overall behavior of \(y = x^{2/5}\).