I have a function:
f(x) =
x^2 - 4x + 1
---------------
x^2 + 1
Now, I'm asked this:
g(x) = bf(x) - 4
g(x)'s absolute maximum is 5.
How to find the variable "b"?
I must mention that I've never learned in school anything about absolute maximum, only regular maximum so if it'll be detailed it'll help a lot in understanding it.
The absolute maximum is the largest value for the function for all the values of x from negative infinity to positive infinity.
If you graph f(x) = [x2 - 4x + 1 ] / [ x2 + 1 ] you will find that the highest value for f(x) occurs at (-1, 3).
Now, the problem is to get the maximum value to occur at (-1, 5).
If you multiply f(x) by 3 [to create g(x) = 3f(x)], the maximum value occrs at (-1, 9).
Now, subtract 4 from this function, creating g(x) = 3f(x) - 4, the maximum value occurs at (-1, 5).
Thus, the value of b is 3.