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What is the maximum value of 4(x + 7)(2 - x) + 4(x + 7)(6 -  x), over all real numbers x?

 Feb 14, 2022
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4 ( x + 7)(2-x)  + 4(x + 7) (6-x)   =

 

4[ 2x + 14 -x^2 - 7x ]  + 4 [ 6x + 42 -x^2 -7x ]  =

 

4 [ -x^2 -5x + 14 ] +  4 [ -x^2 -x + 42 ]

 

-4x^2 -20x + 56  - 4x^2 -4x + 168

 

-8x^2 - 24x  + 224

 

This is a parabola with the x coordinate of the vertex given by  -b/ (2a)  =  24 / ( 2 *-8)  = 24/ -16  =- 1.5

 

And  y coordinate of the  vertex is the max =  -8(-1.5)^2  -24(-1.5) + 224 = -18 + 36 + 224  =    242

 

 

cool cool cool

 Feb 14, 2022

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