Find the zeros of the function [tex]x^3+(3+\sqrt{2}) x^2+4 x+6.7[/tex] whose graph is given below. Round your answer to four decimal places.
Select one:
a. -6.7703
b. -3.8265
c. -2.6267
d. -0.9309
e. -4.9423
Answer (1)
We are given the cubic function f ( x ) = x 3 + ( 3 + 2 ) x 2 + 4 x + 6.7 and asked to find its zeros.
Use the calculate_function_approximate_real_valued_roots tool to find the approximate real-valued roots of the function in the interval [ − 7 , 1 ] .
The tool returns the root − 3.8264548978016317 .
Round the root to four decimal places to get the final answer: − 3.8265 .
Explanation
Understanding the Problem We are given the cubic function f ( x ) = x 3 + ( 3 + 2 ) x 2 + 4 x + 6.7 and asked to find its zeros. The zeros of a function are the x -values where the function equals zero, i.e., the x -intercepts of the graph of the function. We are also given the graph of the function, which helps us visually estimate the zeros. We need to round our answer to four decimal places.
Finding the Root From the graph, we can see that there is one real root. We can use the calculate_function_approximate_real_valued_roots tool to find the approximate real-valued roots of the function. We first try the interval [ − 7 , 1 ] . The tool returns the root − 3.8264548978016317 . We round this to four decimal places to get − 3.8265 .
Confirming the Root To confirm that this is the only real root, we can try other intervals. We try the interval [ − 1 , 0 ] . The tool returns an empty list, which means there are no roots in this interval. We also try the interval [ − 7 , − 3 ] . The tool returns the root − 3.8264548978016313 , which is the same root we found earlier.
Final Answer Therefore, the only real root of the function, rounded to four decimal places, is − 3.8265 .
Examples
Finding the zeros of a function is a fundamental concept in mathematics with numerous real-world applications. For example, in physics, the zeros of a projectile's height function can determine when the projectile hits the ground. In engineering, finding the roots of a system's characteristic equation helps determine the system's stability. In economics, the zeros of a cost function can represent break-even points where revenue equals expenses. Understanding how to find and interpret the zeros of functions is therefore crucial in many fields.
