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Chapter 1: Problem 12

(Graphing program required.) Use technology to graph each function. Then approximate the \(x\) intervals where the function is concave up, and then where it is concave down a. \(f(x)=x^{3}\) b. \(g(x)=x^{3}-4 x\)

Short Answer

Expert verified
a. Concave up: \( x > 0 \), Concave down: \( x < 0 \); b. Concave up: \( x > \frac{2}{3} \), Concave down: \( x < \frac{2}{3} \)

Step by step solution

01

- Graph the Function f(x)

Input the function \( f(x) = x^3 \) into a graphing program or calculator. Observe the shape and curvature of the graph. Identify where the graph is concave up and concave down visually.
02

- Determine Concavity for f(x)

To confirm and approximate where the function is concave up and concave down, calculate the second derivative: \[ f''(x) = \frac{d^2}{dx^2}(x^3) = 6x \] The function is concave up where \( f''(x) > 0 \), which is when \( x > 0 \). The function is concave down where \( f''(x) < 0 \), which is when \( x < 0 \).
03

- Graph the Function g(x)

Input the function \( g(x) = x^3 - 4x \) into the graphing program or calculator. Observe the shape and curvature of the graph. Identify where the graph is concave up and concave down visually.
04

- Determine Concavity for g(x)

To confirm and approximate where the function is concave up and concave down, calculate the second derivative: \[ g''(x) = \frac{d^2}{dx^2}(x^3 - 4x) = 6x - 4 \] The function is concave up where \( g''(x) > 0 \), which is when \( x > \frac{2}{3} \). The function is concave down where \( g''(x) < 0 \), which is when \( x < \frac{2}{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing functions is an essential skill in understanding their behavior and characteristics. When you graph a function, you visualize the relationship between the input values (x) and the output values (y).
For example, input the function \( f(x) = x^3 \) or \( g(x) = x^3 - 4x \) into a graphing calculator or software. Observe the graph's shape and curvature. This step helps in identifying the function's general behavior and aids in detecting gestures like concavity.
Make sure to document these visual observations carefully, as they will guide you through further analysis.
Second Derivative
The second derivative of a function provides insight into its concavity. The process of taking the second derivative involves differentiating the function twice.
For instance, the second derivative of \( f(x) = x^3 \) is: \ f''(x) = 6x \.
Similarly, the second derivative of \( g(x) = x^3 - 4x \) is: \ g''(x) = 6x - 4 \.
These second derivatives will be crucial for determining concavity and interpreting changes in the slope of the function.
Concavity Intervals
Concavity intervals tell us where a function is concave up or concave down. Use the second derivative to find these intervals.
A function is concave up when its second derivative is greater than zero. For \( f(x) = x^3 \), this occurs when \ f''(x) > 0 \ or when \( x > 0 \). For \( g(x) = x^3 - 4x \), it happens when \ g''(x) > 0 \ or \( x > \frac{2}{3} \).
Conversely, a function is concave down when its second derivative is less than zero. For \( f(x) = x^3 \), this is when \ f''(x) < 0 \ or \( x < 0 \). For \( g(x) = x^3 - 4x \), it occurs when \ g''(x) < 0 \ or when \( x < \frac{2}{3} \).
These intervals help in understanding how the function curves at different sections along the x-axis.
Function Analysis
Analyzing a function involves looking at several aspects, including its graph, first and second derivatives, and intervals of concavity.
Here's a quick way to perform a function analysis:
  • Graph the function to get a visual overview.
  • Calculate the first and second derivatives.
  • Determine where the first derivative (the slope) changes sign. These points can suggest local maxima or minima.
  • Find where the second derivative is positive or negative to identify concavity intervals.

Through this structured analysis, you gain a comprehensive understanding of the function's behavior.

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Most popular questions from this chapter

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