+1839 If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4 +5t^2 - 4t + 18$?
+9673 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.
+9673 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.
