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Find the largest real number $c$ such that $1$ is in the range of $f(x)=x^2-5x+c+x^2-8x$.

 Aug 27, 2023
 #1
avatar+128732 
+1

Simplifying  f(x)  we have

 

2x^2 -13x + c

 

This will be a parabola that turns  upward.....we can let the vertex lie on the line y =1

 

The x coordinate of the  vertex is  (13 / 4)

 

And we  want the function  to =  1    ......so......

 

2(13/4)^2  - 13(13/4)  + c  = 1

 

2(169) / 16   - 169 / 4  + c  = 1

 

169 / 8  - 169/ 4 +  c  = 1

 

-169/ 8  + c  = 1

 

c =  1 + 169/8

 

c =  8/8 + 169/8

 

c = 177 / 8   =   22.125

 

cool cool cool

 Aug 27, 2023

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