Inequality Basic

Problem - 3220
Let $a, b$ be positive numbers, show that $$\frac{1}{2}(a+b)+\frac{1}{4}\ge \sqrt{\frac{a+b}{2}}$$

Show solution
$$\frac{1}{2}(a+b)+\frac{1}{4} - \sqrt{\frac{a+b}{2}}=\Big(\frac{1}{2}-\sqrt{\frac{a+b}{2}}\Big)^2\ge 0$$

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