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If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4 +5t^2 - 4t + 18$?

 Apr 9, 2024

Best Answer 

 #1
avatar+9664 
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I assume you mean minimum.

 

Simplifying and completing the square,

 

\(-t^2 +8t-4+5t^2-4t+18\\ =4t^2 +4t + 14\\ =4t^2 +4t+1 +13\\ =(2t+1)^2 + 13\\ \)

Note that \((2t + 1)^2 \geq 0\) for all real t. Then the minimum is 13. 

 

The expression can be arbitrarily large so there is no maximum.

 Apr 9, 2024
 #1
avatar+9664 
+1
Best Answer

I assume you mean minimum.

 

Simplifying and completing the square,

 

\(-t^2 +8t-4+5t^2-4t+18\\ =4t^2 +4t + 14\\ =4t^2 +4t+1 +13\\ =(2t+1)^2 + 13\\ \)

Note that \((2t + 1)^2 \geq 0\) for all real t. Then the minimum is 13. 

 

The expression can be arbitrarily large so there is no maximum.

MaxWong Apr 9, 2024

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