Problem 18 - Thomas Calculus

Question

Graph the functions in Exercises \(7-18 .\) What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$ y=-x^{2 / 3} $$

Step-by-Step Solution

Answer

The graph is symmetric about the y-axis, increasing on \((-\infty,0)\) and decreasing on \((0,\infty)\).

Step 1: Identify the Function Type

The function given is a cubic root function, specifically, \( y = -x^{2/3} \). This means it is not a polynomial with an even power, which would affect the kind of symmetry it may have.

Step 2: Check for Symmetries

To determine symmetry, check for even or odd functions. An even function satisfies \( f(-x) = f(x) \) and an odd function satisfies \( f(-x) = -f(x) \). Compute \( f(-x) = -(-x)^{2/3} = -x^{2/3} = f(x) \), indicating that the function is even and symmetric about the y-axis.

Step 3: Determine Critical Points and Sign Changes

The derivative of the function needs to be evaluated to find intervals of increase or decrease. Compute the first derivative \( f'(x) = -\frac{2}{3}x^{-1/3} \). The derivative changes signs due to \( x^{-1/3} \) when crossing \( x=0 \).

Step 4: Analyze Intervals of Increase/Decrease

Set \( f'(x) \) to greater or less than zero to understand where the function increases or decreases. From \( f'(x) \): - When \( x < 0 \), \( f'(x) > 0 \), indicating the function is increasing.- When \( x > 0 \), \( f'(x) < 0 \), indicating the function is decreasing.

Step 5: Conclusion from Analysis

The graph of \( y=-x^{2/3} \) is symmetric about the y-axis. The function is increasing on the interval \( (-\infty, 0) \) and decreasing on the interval \( (0, \infty) \).

Key Concepts

Cubic Root Function

A cubic root function typically involves the cube root of an expression, but in this case, we have a different variant in the form of \( y = -x^{2/3} \). This expression means that the function is defined by taking \( x \) to the power of \( \frac{2}{3} \), introducing a cubic root component.
What makes this function unique is that it combines both a square and a cube root aspect. This structure results in intriguing behaviors, such as influencing the graph's shape and symmetry characteristics.
Additionally, in functions like \( y = -x^{2/3} \), the negative sign in front of the power means the graph will reflect over the x-axis, further adding to its distinct form.

Intervals of Increase and Decrease

To find where a function increases or decreases, we look at the derivative, which gives the slope of the tangent at any point of the function. For our function \( y = -x^{2/3} \), the derivative is \( f'(x) = -\frac{2}{3}x^{-1/3} \).
This derivative tells us how the rate of change in the function behaves. When the derivative is positive, the function increases. Conversely, if it’s negative, the function decreases.
For \( y = -x^{2/3} \):

• When \( x < 0 \), \( f'(x) > 0 \). Here, the function increases as we move towards zero from the left.
• When \( x > 0 \), \( f'(x) < 0 \). In this interval, the function decreases as we move away from zero towards positive infinity.

Even Function Symmetry

An even function shows symmetry about the y-axis, meaning the left and right sides of the graph mirror each other. For a function to be even, it satisfies the condition \( f(-x) = f(x) \).
Analyzing \( y = -x^{2/3} \), we compute \( f(-x) = -(-x)^{2/3} \), which simplifies to \( -x^{2/3} = f(x) \). This confirms the function is even.
Thus, for \( y = -x^{2/3} \), the graph is symmetric about the y-axis. This symmetry simplifies understanding its geometric behavior and helps in predicting its reflection attributes.

Derivative Analysis

Derivative analysis is crucial for understanding the behavior of functions, especially regarding their increasing and decreasing nature. For the cubic root function \( y = -x^{2/3} \), the derivative \( f'(x) = -\frac{2}{3}x^{-1/3} \) provides essential insights.
By setting \( f'(x) = 0 \), we identify critical points, although for this function, it's undefined at \( x = 0 \) due to natural division by zero issues from \( x^{-1/3} \).
Nevertheless, analyzing the derivative's sign changes across the x-axis reveals:

• Positive derivative for \( x < 0 \), indicating an increasing segment.
• Negative derivative for \( x > 0 \), indicating a decreasing segment.

Such detailed derivative analysis allows us to draw valid conclusions about the graph's behavior without confusion.