Question

Știind că este x este număr real, determinați valoarea maximă a expresiei:
a) -x^2-4x-4
b) -x^2+2x-3
c) -x^2-2x+5
d) -x^2+8x

Answer 1

 

$\displaystyle\bf\\\text{Deoarece toate expresiile au coeficientul lui }~x^2~\text{negativ, au un maxim.}\\\\ a)\\max(-x^2-4x-4)=\frac{-\Delta}{4a}=\frac{-(b^2-4ac)}{4a}=\\\\=\frac{-((-4)^2-4\cdot(-1)\cdot(-4))}{4\cdot(-1)}=\frac{-(16-16)}{-4}=\frac{-0}{-4}=\boxed{\bf0}\\\\b)\\max(-x^2+2x-3)=\frac{-\Delta}{4a}=\frac{-(b^2-4ac)}{4a}=\\\\=\frac{-(2^2-4\cdot(-1)\cdot(-3))}{4\cdot(-1)}=\frac{-(4-12)}{-4}=\frac{-(-8)}{-4}=\frac{8}{-4}=\boxed{\bf-2}$

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$\displaystyle\bf\\c)\\max(-x^2-2x+5)=\frac{-\Delta}{4a}=\frac{-(b^2-4ac)}{4a}=\\\\=\frac{-((-2)^2-4\cdot(-1)\cdot5)}{4\cdot(-1)}=\frac{-(4+20)}{-4}=\frac{-24}{-4}=\frac{24}{4}=\boxed{\bf6}\\\\d)\\max(-x^2+8x)=\frac{-\Delta}{4a}=\frac{-(b^2-4ac)}{4a}=\\\\=\frac{-(8^2-4\cdot(-1)\cdot0)}{4\cdot(-1)}=\frac{-(64-0}{-4}=\frac{-64}{-4}=\frac{64}{4}=\boxed{\bf16}$