SOLUTION: Given f(1-x)+(1-x)f(x)=5 for all real number x, find the maximum value that is attained by f(x). Now, this made me think really deep, but I just can't seem to analyze it. Please

Question 1048554: Given f(1-x)+(1-x)f(x)=5 for all real number x, find the maximum value that is attained by f(x).
Now, this made me think really deep, but I just can't seem to analyze it.
Please help, thanks.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!

====> Replacing x by 1-x, we get
Using Cramer's rule to find f(x),

.
(As a check, , which is the expression also obtained when x is replaced by 1-x in the preceding expression for f(x).)

===> f'(x) =

Setting this derivative equal to 0, we get the critical values x = 0, 2.
By using the first derivative test, we find a local max at x = 0, and a local minimum at x = 2.

At x = 0, f(0) = 5.
At x = 2, f(2) = -5/3.

Therefore, the absolute max value of f(x) is 5, and the absolute min value of f(x) is -5/3.
We confirm by graphing f(x):

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