$f(x)=2 x^3-8 x^2-2 x+5$
Answer (2)
The cubic function f ( x ) = 2 x 3 − 8 x 2 − 2 x + 5 can be solved to find its roots. The approximate real roots are x → − 0.830 , 0.735 , 4.095 . Numerical methods can be used for solving such cubic equations due to their complexity.
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The problem provides the cubic function f ( x ) = 2 x 3 − 8 x 2 − 2 x + 5 .
We aim to find the real roots of the function, i.e., solve f ( x ) = 0 .
Using numerical methods, we approximate the real roots.
The approximate real roots are x ≈ − 0.830 , 0.735 , and 4.095 , so the final answer is − 0.830 , 0.735 , 4.095 .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 x 3 − 8 x 2 − 2 x + 5 . The problem does not specify what we need to do with this function. Let's assume we want to find the real roots of the function, i.e., the values of x for which f ( x ) = 0 .
Finding the Roots To find the roots of the function f ( x ) = 2 x 3 − 8 x 2 − 2 x + 5 , we need to solve the equation 2 x 3 − 8 x 2 − 2 x + 5 = 0 . This is a cubic equation, and finding its roots analytically can be complex. However, we can use numerical methods or tools to approximate the roots.
Approximate Real Roots Using a numerical method, we find the approximate real roots of the equation 2 x 3 − 8 x 2 − 2 x + 5 = 0 to be approximately x 1 ≈ − 0.830 , x 2 ≈ 0.735 , and x 3 ≈ 4.095 .
Final Answer Therefore, the approximate real roots of the function f ( x ) = 2 x 3 − 8 x 2 − 2 x + 5 are x ≈ − 0.830 , 0.735 , 4.095 .
Examples
Cubic functions like the one in this problem are used in various fields, such as physics to model projectile motion or in economics to represent cost functions. Finding the roots of such functions can help determine key points like the launch angle needed to hit a target or the production level that minimizes cost. Understanding how to analyze and solve polynomial equations is crucial for making informed decisions in these real-world scenarios.
