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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?

 Feb 21, 2021
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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

 Feb 21, 2021
edited by ElectricPavlov  Feb 21, 2021

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