Problem 41 - Calculus: Early Transcendentals

Question

Describe the concavity of the functions below. $$ y=3 x^{2}-\left(1 / x^{2}\right) $$

Step-by-Step Solution

Answer

Concave up: \((-\infty, -1) \cup (1, \infty)\); Concave down: \((-1, 0) \cup (0, 1)\).

Step 1: Find the Second Derivative

To determine the concavity of the function, we first find the second derivative. The function is \( y = 3x^2 - \frac{1}{x^2} \). Begin by finding the first derivative: \( y' = 6x + \frac{2}{x^3} \). Now, differentiate \( y' \) with respect to \( x \) to find \( y'' \). \( y'' = 6 - \frac{6}{x^4} \).

Step 2: Analyze the Second Derivative

The second derivative \( y'' = 6 - \frac{6}{x^4} \) tells us about the concavity. Analyze \( y'' \) to determine the intervals of concavity. When \( y'' > 0 \), the graph is concave up; when \( y'' < 0 \), the graph is concave down.

Step 3: Set the Second Derivative Greater Than Zero

Set \( y'' > 0 \) to find where the function is concave up. \[6 - \frac{6}{x^4} > 0\]Rearranging gives \( x^4 > 1 \). Thus, \( x > 1 \) or \( x < -1 \).

Step 4: Set the Second Derivative Less Than Zero

Set \( y'' < 0 \) to find where the function is concave down. \[6 - \frac{6}{x^4} < 0\]Rearranging gives \( x^4 < 1 \). Thus, \(-1 < x < 1 \), but since \( x eq 0 \), this interval is \(-1 < x < 0 \) and \( 0 < x < 1 \).

Step 5: Conclusion on Concavity

Based on the analysis, the function \( y = 3x^2 - \frac{1}{x^2} \) is concave up on the intervals \((-\u221e, -1) \cup (1, \infty)\) and concave down on \((-1, 0) \cup (0, 1)\).

Key Concepts

Second Derivative

The second derivative of a function is a powerful tool in calculus that helps us understand the curvature or concavity of a graph. Simply put, it tells us about the "bending" of the graph. If you start with a function, the first derivative can tell you the slope or the rate of change at any given point. By taking one more derivative, you get the second derivative, which provides insight into how that slope itself is changing.

For the function given in our exercise, which is \( y = 3x^2 - \frac{1}{x^2} \), the first step was to compute its first derivative, \( y' = 6x + \frac{2}{x^3} \). Then, by differentiating once more, we obtain the second derivative: \( y'' = 6 - \frac{6}{x^4} \).

This second derivative can now help us to determine the concavity of the function across different intervals.

Concave Up

When the second derivative of a function is positive over an interval, the graph of the function is said to be concave up on that interval. This means that the graph looks like an upward-opening bowl or the shape of a U.

Mathematically, this is expressed as \( y'' > 0 \). For the function \( y = 3x^2 - \frac{1}{x^2} \), to find where it is concave up, we set up the inequality \( 6 - \frac{6}{x^4} > 0 \).

By solving this inequality—\( x^4 > 1 \)—we find that the function is concave up in the intervals \( x > 1 \) and \( x < -1 \). Hence, beyond these ranges, the curve bends upwards.

Concave Down

Conversely, when the second derivative is negative over an interval, the graph is concave down. This appearance resembles a downward-opening bowl or the shape of an upside-down U.

For the function in question, \( y'' < 0 \) describes the concave down regions. The setup is: \( 6 - \frac{6}{x^4} < 0 \), leading to the inequality \( x^4 < 1 \).

The graph is concave down in the intervals \(-1 < x < 0\) and \(0 < x < 1\), where it bends downwards. It's important to note that \( x = 0 \) is not included in these intervals, as the original function is undefined at \( x = 0 \).

Intervals of Concavity

Intervals of concavity are specific sections of the graph where the curvature is consistent—either concave up or concave down. By inspecting the second derivative, we identify these intervals and can describe how the graph behaves in different regions.

For the function \( y = 3x^2 - \frac{1}{x^2} \), our analysis shows:

• It is concave up on the intervals \((−\infty,−1)\) and \((1, \infty)\).
• It is concave down on the intervals \((-1,0)\) and \((0,1)\).

Understanding these intervals of concavity helps us visualize the graph effectively, knowing where it bends upwards or downwards based on the value of \( x \). Such analysis is critical in sketching the graph of the function and predicting its behavior in response to changes in \( x \).