f(x) = (x^2 + 3)/(4x + 1) x != -1/4 find the set of values for which f(x) is increasing.

First we have to find the first derivative of f(x), we do this using the quotient rule. Once we have obtained the first derivative, we know this gives us the rate of change of f(x), we can therefore say that the function f(x) is increasing for values of x which give a positive number when substituted into this first derivative. We look for the turning points of f(x) by equating the first derivative to zero and finding the values of x for which that is true. This is a simple quadratic which can be solved by inspection or using the quadratic formula if necessary. These values of x give us the turning points in f(x) but we still need to know if f(x) increases inside or outside of this range, we can do this by substituting in a value of x between these limits into the first derivative equation we found previously. In this case the value between the roots would be negative, so the function increases outside these values, this last step can also be done graphically.