$f(x)=x^6-4 x^4+3 x^2-10$

We analyzed the function f ( x ) = x 6 − 4 x 4 + 3 x 2 − 10 by finding its first and second derivatives, critical points, and estimating its real roots to be approximately − 1.972 and 1.972 . This analysis is essential for understanding the behavior of polynomial functions in mathematics as they relate to optimization and modeling in various disciplines.
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Answered by Anonymous | 2025-07-04

Find the first derivative: f ′ ( x ) = 6 x 5 − 16 x 3 + 6 x .
Find the second derivative: f ′′ ( x ) = 30 x 4 − 48 x 2 + 6 .
Find the real roots of f ( x ) = 0 : x ≈ − 1.972 and x ≈ 1.972 .
The approximate real roots of the function are − 1.972 , 1.972 ​ .

Explanation

Problem Analysis We are given the function f ( x ) = x 6 − 4 x 4 + 3 x 2 − 10 . Our goal is to analyze this function by finding its roots, critical points, and concavity.

Finding the First Derivative First, let's find the first derivative of the function, f ′ ( x ) . Using the power rule, we have: f ′ ( x ) = 6 x 5 − 16 x 3 + 6 x

Finding the Second Derivative Now, let's find the second derivative of the function, f ′′ ( x ) . Again, using the power rule, we have: f ′′ ( x ) = 30 x 4 − 48 x 2 + 6

Finding Critical Points To find the critical points, we set f ′ ( x ) = 0 :
6 x 5 − 16 x 3 + 6 x = 0 2 x ( 3 x 4 − 8 x 2 + 3 ) = 0 So, x = 0 is one critical point. To find other critical points, we need to solve 3 x 4 − 8 x 2 + 3 = 0 . Let y = x 2 , then we have 3 y 2 − 8 y + 3 = 0 . Using the quadratic formula: y = 2 ( 3 ) − ( − 8 ) ± ( − 8 ) 2 − 4 ( 3 ) ( 3 ) ​ ​ = 6 8 ± 64 − 36 ​ ​ = 6 8 ± 28 ​ ​ = 6 8 ± 2 7 ​ ​ = 3 4 ± 7 ​ ​ So, x 2 = 3 4 ± 7 ​ ​ , which gives us x = ± 3 4 + 7 ​ ​ ​ and x = ± 3 4 − 7 ​ ​ ​ .

Finding the Roots To find the roots of the function, we set f ( x ) = 0 :
x 6 − 4 x 4 + 3 x 2 − 10 = 0 Using a calculator, we find the approximate real roots to be x ≈ − 1.972 and x ≈ 1.972 .

Analyzing Concavity To determine the concavity, we analyze the sign of f ′′ ( x ) . We set f ′′ ( x ) = 0 :
30 x 4 − 48 x 2 + 6 = 0 5 x 4 − 8 x 2 + 1 = 0 Let z = x 2 , then 5 z 2 − 8 z + 1 = 0 . Using the quadratic formula: z = 2 ( 5 ) − ( − 8 ) ± ( − 8 ) 2 − 4 ( 5 ) ( 1 ) ​ ​ = 10 8 ± 64 − 20 ​ ​ = 10 8 ± 44 ​ ​ = 10 8 ± 2 11 ​ ​ = 5 4 ± 11 ​ ​ So, x 2 = 5 4 ± 11 ​ ​ , which gives us x = ± 5 4 + 11 ​ ​ ​ and x = ± 5 4 − 11 ​ ​ ​ .

Final Answer The approximate real roots of the function are x ≈ − 1.972 and x ≈ 1.972 .


Examples
Understanding the behavior of polynomial functions like f ( x ) = x 6 − 4 x 4 + 3 x 2 − 10 is crucial in many fields. For example, in engineering, analyzing such functions can help in designing stable structures or predicting the behavior of systems. In economics, these functions can model cost curves or revenue streams, aiding in decision-making. Furthermore, in computer graphics, polynomial functions are used to create smooth curves and surfaces, enhancing visual realism. By understanding roots, critical points, and concavity, we can effectively optimize designs, predict outcomes, and create realistic models.

Answered by GinnyAnswer | 2025-07-04