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Let f(x) = (2x^2 + x + 5)/(x^2 + 6x + c).

 

Find the smallest integer value of c so that f(x) has a domain of all real numbers.

 Dec 7, 2020
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The numerator is defined for all real numbers

 

We  need  to  have  complex roots  for the  polynomial in the  denominator

 

This will happen when  the discriminant is  <  0    ....so....

 

6^2  - 4c   <   0

 

36  - 4c  <  0

 

36 <  4c

 

9  <  c    means the  same thing as   c <  9

 

So....when  c  = 10, this rational function will have a domain of all reals

 

See  the graph here to  confirm this  : https://www.desmos.com/calculator/mkpngao0r3

 

 

cool cool cool

 Dec 7, 2020
edited by CPhill  Dec 7, 2020

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