a) Xét hiệu : VT - VP
= \(\dfrac{\left(a+b\right)^2}{4}\) _ ab = \(\dfrac{a^2+2ab+b^2}{4}\)- \(\dfrac{4ab}{4}\)
= \(\dfrac{a^2-2ab+b^2}{4}\) = \(\dfrac{\left(a-b\right)^2}{4}\)
Có : (a - b )2 \(\ge\) 0 => \(\dfrac{\left(a-b\right)^2}{4}\) \(\ge\) 0 .
(bất phương trình đúng ) .
=> VT - VP \(\ge\) 0 => ( \(\dfrac{a+b}{2}\))2 \(\ge\) ab .
b) Xét hiệu ; VP - VT
= \(\dfrac{a^2+b^2}{2}\)-(\(\dfrac{a+b}{2}\))2
= \(\dfrac{2a^2+2b^2-\left(a^2+2ab+b^2\right)}{4}\)
= \(\dfrac{\left(a-b\right)^2}{4}\) .
Có : (a-b)2 \(\ge\) 0 => \(\dfrac{\left(a-b\right)^2}{4}\) \(\ge\) 0 .
VP - VT \(\ge\) 0 .
Vậy ( \(\dfrac{a+b}{2}\) )2 \(\le\) \(\dfrac{a^2+b^2}{2}\) .
