MathNet 2016-part-02 — 184 problems (EN–VI) Question 31

Let $\lambda$ be a nonnegative real number and such that $\frac{a+b}{2} \ge \lambda\sqrt{ab}+(1-\lambda)\sqrt{\frac{a^2+b^2}{2}}$ holds for all positive real numbers $a$ and $b$. Find the smallest possible value of $\lambda$.