What is the smallest integer value of c such that the function f(x) = (2x^2 + x + 5)/(x^2 + 14x + c) has a domain of all real numbers?
The denominator is a bowl shaped parabola (due to the POSITIVE coefficient for x2).... all values of this parabola must be > 0 for the domain to be all real numbers....
if the denominator touches 0 or turns negative (by crossing 0) the domain will be limited....
x^2 + 14x + c > 0 the vertex (the low point of the parabola) of this parabola will occur at - b/2a = -14/2 = -7
at this point the parabola must be >0
(-7)^2 + 14 (-7) + c > 0
c > 49 so the smallest integer c would be 50