Real Roots of Polynomial Minimization Problem

In summary, for an integer $n\ge 2$, the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value at $x=\dfrac{n+1}{2}$, which is the unique global minimum point. The derivative $f'(x)$ is an increasing function and has only one zero, making $x=\dfrac{n+1}{2}$ the only critical point.
  • #1
anemone
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For an integer $n\ge 2$, find all real numbers $x$ for which the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value.
 
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anemone said:
For an integer $n\ge 2$, find all real numbers $x$ for which the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value.
[sp]
Since $x=\dfrac{n+1}{2}$ is an axis of symmetry, the point $x=\dfrac{n+1}{2}$ is either a minimum of a maximum, depending on the shape of the quartic.

However, the derivative $f'(x) = 4\left((x-1)^3+\cdots+(x-n)^3\right)$ is an increasing function (since it is a sum of increasing functions). Therefore, $f'(x)$ as exactly one zero, and $f(x)$ has only one critical point.

We conclude that $x=\dfrac{n+1}{2}$ is the unique global minimum.
[/sp]
 

FAQ: Real Roots of Polynomial Minimization Problem

What is a "Real Roots of Polynomial Minimization Problem"?

A "Real Roots of Polynomial Minimization Problem" is a mathematical problem in which the goal is to find the minimum value of a polynomial function with real coefficients. This involves finding the values of the independent variables that result in the lowest possible output value for the function.

Why is finding the real roots of a polynomial minimization problem important?

Finding the real roots of a polynomial minimization problem is important because it allows us to determine the minimum value of the polynomial function, which can have many real-world applications. For example, it can help us optimize processes in engineering, economics, and other fields where minimizing a function is crucial.

What methods are commonly used to solve real roots of polynomial minimization problems?

The most common methods used to solve real roots of polynomial minimization problems are the derivative method, the quadratic formula, and the Newton-Raphson method. These methods involve finding the critical points of the polynomial function and determining which one results in the minimum value.

Can a polynomial function have more than one minimum value?

Yes, a polynomial function can have more than one minimum value. This occurs when the polynomial has multiple critical points, and each one results in a minimum value. In this case, the global minimum is the lowest of all the minimum values.

Are there any limitations to finding the real roots of polynomial minimization problems?

Yes, there are limitations to finding the real roots of polynomial minimization problems. One limitation is that the polynomial function must have real coefficients. Additionally, some polynomial functions may not have a minimum value, making it impossible to solve the minimization problem.

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