Maximizing a 3rd Order Polynomial: Finding Duplicate Solutions

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SUMMARY

The discussion centers on the function f(x) = (x+4)^2(a-x) and the conditions under which it has two solutions for f(x) = T, where T is greater than 0. A key conclusion is that a 3rd order polynomial can have two solutions only if one of them is a duplicate, which occurs at an extreme value. To find this extreme value, participants emphasize the necessity of taking the derivative of the function. The cubic formula is mentioned but not recommended for this problem.

PREREQUISITES
  • Understanding of 3rd order polynomials
  • Knowledge of calculus, specifically derivatives
  • Familiarity with extreme values in functions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the process of taking derivatives of polynomial functions
  • Learn about finding extreme values and their significance
  • Explore the implications of duplicate solutions in polynomial equations
  • Investigate alternative methods to the cubic formula for solving polynomials
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Mathematicians, students studying calculus, and anyone interested in polynomial functions and their properties will benefit from this discussion.

Bushy
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Consider the function

f(x) = (x+4)^2(a-x)

Let T be greater than 0, find 'a' such that f(x) = T has 2 solutions.

No idea how to kick this one off..
 
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Exactly how badly do you need the solution? This is basically a generalization of your previous thread, and that one was already answered by MarkFL, to your dissatisfaction. I don't immediately see another solution, unless you want to use the cubic formula (which I really don't recommend).

(Bandit)
 
Bushy said:
Consider the function

f(x) = (x+4)^2(a-x)

Let T be greater than 0, find 'a' such that f(x) = T has 2 solutions.

No idea how to kick this one off..

Hi Bushy!

A 3rd order polynomial has only 2 solutions if one of the 2 is a duplicate solution, meaning it has an extreme value there.
To find an extreme value you're supposed to take the derivative...
 

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