MHB Maximizing a 3rd Order Polynomial: Finding Duplicate Solutions

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To maximize the 3rd order polynomial f(x) = (x+4)^2(a-x) and find 'a' such that f(x) = T has two solutions, one must identify a duplicate solution at an extreme value. This requires taking the derivative of the function to locate critical points. The discussion emphasizes that a cubic function can only yield two solutions if one is a repeated root, which corresponds to an extremum. The cubic formula is mentioned but not recommended for this problem. Understanding the relationship between the function's derivative and its critical points is essential for solving the equation.
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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...
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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