This problem might as well be stated:

"If the capital of Georgia is Atlanta, then prove the equation 
xlogx = 3-x is satisfied by at least one value of x lying 
between 1 and 3."

That's because "f(x) = (x-3)logx" has no more to do with whether

"the equation xlogx = 3-x is satisfied by at least one value of x 
lying between 1 and 3" than the capital of Georgia being Atlanta 
has to do with it!

Thus we ignore the f(x) part and take the problem to be:

Prove that the equation 

 

is satisfied by at least one value of x lying between 1 and 3.

Consider the function which is continuous on [1,3]







g(1) is negative and g(3) is positive.

Thus there is some number, say h, on [1,3] such that



Thus



and



[Note: If you require that h be an algebraic number then
choose h to be ]

Edwin