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

Algebra ->  Functions -> 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      Log On


   



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) About Me  (Show Source):
You can put this solution on YOUR website!
f%281-x%29%2B%281-x%29f%28x%29=5
====> Replacing x by 1-x, we get f%28x%29%2Bxf%281+-+x%29=5
Using Cramer's rule to find f(x),

.
(As a check, f%281-x%29+=+%28-5x%29%2F%28-x%5E2%2Bx-1%29, which is the expression also obtained when x is replaced by 1-x in the preceding expression for f(x).)

===> f'(x) = %285x%5E2-10x%29%2F%28-x%5E2+%2B+x-1%29%5E2

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):
graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+%285x-5%29%2F%28-x%5E2%2Bx-1%29+%29