Question

Dau 50 puncte + coroana. (ex. 29 si 30)

Answer 1

29.
$\it \sqrt{a^{2} - 2 \cdot 2 \cdot a + 4 + 16} + \sqrt{b^{2} + 2 \cdot 8 \cdot b + 64 + 4} = \sqrt{(a-2)^{2} + 16} + \sqrt{(b+8)^{2} + 4} \\ \\ (a-2)^{2} \geq 0 , (b+8)^{2} \geq 0 \\ Min \sqrt{(a-2)^{2} + 16} = \sqrt{0^{2} +16} = \sqrt{16} = 4 \\ \\ \\ Min \sqrt{(b+8)^{2}+4} = \sqrt{0^{2} + 4} = \sqrt{4} = 2 \\ \\ Min_expresie = 4 +2=6$
30.
$20 - \sqrt{a^{2} -10a+50} = 20 - \sqrt{(a^{2} - 2 \cdot 5 \cdot a + 25) + 25} =20 - \sqrt{(a-5)^{2} + 25} \\ \\ Pt. \ a \ obtine \ maximul \ expresiei \ date \ , \ radicalul \ trebuie \ sa \ fie \ mai \ mic \ decat \ 20 , \ (a-5)^{2} \geq 0 \Rightarrow \sqrt{(a-5)^{2} + 25} = \sqrt{0^{2} +25} = 5 \\ \\ Max_expresie= 20-5 = 15$