Question

Докажите, что если a>0 и b>0, то
(a+2)×(b+32)×(12,5+2/ab)>320

Answer 1

По AM>=GM

(a+2)(b+32)*(25/2+2/ab) >= (a+2)(b+32)*2*√(25/2*2/ab) = (a+2)(b+32)*10/√ab = (ab+32a+2b+64)*10/√ab >=  (ab+16*√ab+64)*10/√ab =  10*√ab + 640/√ab + 160  >= 2*√(10*√ab * 640/√ab)  + 160 = 2*80+160 = 320