+0  
 
-1
33
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avatar+817 

Find the smallest integer $c$ such that the domain of the function $f(x)=\frac{x^2+1}{x^2-x+c+3x^2-7x}$ is all real numbers.

 Aug 22, 2023
 #1
avatar+129657 
+1

The denominator  cannot  = 0 

 

Simplifying the  denominator  we get

 

4x^2 -8x + c    →   this is an upward turning parabola

 

To have a domain of all  real numbers, this parabola must lie above the  x  axis

 

So.....the discrimiant  must  be <  0  ....so....

 

(-8)^2 -4(4) ( c) <  0

 

64  - 16c <  0

 

64 < 16c

 

4 < c

 

So

 

c >  4

 

The smallest integer  is  5

 

cool cool cool

 Aug 22, 2023

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