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What is the smallest integer value of c such that the function f(x) = (x^2 + 1)/(x^2 - 4x + c)  has a domain of all real numbers?

 Jul 13, 2021

Best Answer 

 #1
avatar+208 
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the only way the expression \(\frac{x^2\:+\:1}{x^2\:-\:4x\:+\:c}\) is not real is if the denominator equals zero

 \(x^2\:-\:4x\:+\:c=0 \)

discriminant is:

 \((-4)^2-4c \\=16-4c\)

to have no real solutions:

\(16-4c<0 \\4c>16 \\c>4\)

smallest integer value is c=5

 

JP

 Jul 14, 2021
 #1
avatar+208 
+1
Best Answer

the only way the expression \(\frac{x^2\:+\:1}{x^2\:-\:4x\:+\:c}\) is not real is if the denominator equals zero

 \(x^2\:-\:4x\:+\:c=0 \)

discriminant is:

 \((-4)^2-4c \\=16-4c\)

to have no real solutions:

\(16-4c<0 \\4c>16 \\c>4\)

smallest integer value is c=5

 

JP

JKP1234567890 Jul 14, 2021

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