Using only the values given in the table for the function [tex]f(x)=-x^3+4 x+3[/tex], what is the largest interval of [tex]x[/tex] values where the function is increasing?
([tex]\square[/tex] , [tex]\square[/tex])
Answer (2)
Find the derivative of the function: f ′ ( x ) = − 3 x 2 + 4 .
Set the derivative to zero to find critical points: x = ± 3 2 ≈ ± 1.1547 .
Determine the interval where the derivative is positive, indicating the function is increasing: ( − 3 2 , 3 2 ) .
Approximate the interval: ( − 1.15 , 1.15 ) .
Explanation
Problem Analysis We are given the function f ( x ) = − x 3 + 4 x + 3 and asked to find the largest interval where the function is increasing, using only the values given in a table (which is not provided).
Find the Derivative To find where the function is increasing, we need to determine where its derivative is positive. The derivative of f ( x ) is f ′ ( x ) = − 3 x 2 + 4 .
Find Critical Points To find the critical points, we set f ′ ( x ) = 0 , which gives − 3 x 2 + 4 = 0 . Solving for x , we get x 2 = 3 4 , so x = ± 3 4 = ± 3 2 ≈ ± 1.1547 .
Determine Increasing Interval The function is increasing where 0"> f ′ ( x ) > 0 . This means 0"> − 3 x 2 + 4 > 0 , which implies 3x^2"> 4 > 3 x 2 , so x 2 < 3 4 . This implies − 3 2 < x < 3 2 . Therefore, the function is increasing on the interval ( − 3 2 , 3 2 ) ≈ ( − 1.1547 , 1.1547 ) .
Apply to Sample Data Since we are restricted to using values from a table that is not provided, we will use a sample table to illustrate the process. Let's assume the table contains the following data points:
( − 2 , − 3 ) , ( − 1.5 , − 2.125 ) , ( − 1 , − 2 ) , ( − 0.5 , 0.375 ) , ( 0 , 3 ) , ( 0.5 , 5.375 ) , ( 1 , 6 ) , ( 1.5 , 5.625 ) , ( 2 , 3 )
We look for consecutive intervals where the y -values are increasing. From x = − 2 to x = 1 , the y -values are generally increasing. After x = 1 , the y -values start to decrease. Therefore, based on this sample data, the largest interval where the function is increasing is ( − 2 , 1 ) .
Final Answer Based on the derivative analysis, the function is increasing on the interval ( − 3 2 , 3 2 ) . However, without the actual table, we can only provide an example using a sample table. The question asks for the largest interval of x values where the function is increasing. Based on the sample data, the function is increasing on the interval ( − 2 , 1 ) .
Final Interval Without the actual table, we can only provide the interval based on the derivative, which is ( − 3 2 , 3 2 ) . Since the question requires an interval ( □ , □ ) , and we don't have the table, we will use the interval derived from the derivative. Approximating the values, we get ( − 1.15 , 1.15 ) .
Examples
Understanding where a function is increasing or decreasing is crucial in many real-world applications. For example, in economics, you might analyze the profit function of a company. Identifying the interval where the profit function is increasing helps determine the production level that maximizes profit. Similarly, in physics, understanding the increasing or decreasing nature of a velocity function helps analyze the acceleration of an object. These concepts are also used in engineering to optimize designs and improve efficiency.
The function f ( x ) = − x 3 + 4 x + 3 is increasing on the interval ( − 3 2 , 3 2 ) ≈ ( − 1.15 , 1.15 ) . This conclusion is drawn from the analysis of the function's derivative. Without additional table data, this is the largest interval identified from the derivative.
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