what's the maximum value of the algebraic expression? 
Suppose that $a,b,c>0$ and satisfy the equation
  $$
a^2+b^2+4c^2=1,
$$
  then what's the maximum of $F(a,b,c)=ab+2ac+3\sqrt{2}bc$ ?

**
My way: $F(a,b,c)=ab+\sqrt{2}bc+2(ac+\sqrt{2}bc)=\cdots$

Use AM-GM inequality we have
$$\dfrac{\sqrt{2}}{2}a^2+\dfrac{\sqrt{2}}{4}b^2\ge ab\tag{1}$$
$$\sqrt{2}c^2+\dfrac{\sqrt{2}}{2}a^2\ge ac\tag{2}$$
$$\dfrac{3\sqrt{2}}{4}b^2+3\sqrt{2}c^2\ge 3\sqrt{2}bc\tag{3}$$
$(1)+(2)+(3)$ we have 
$$ab+ac+3\sqrt{2}bc\le \sqrt{2}(a^2+b^2+4c^2)=\sqrt{2}$$
