Hint:
As $x^4,1/x^2>0$ using Weighted Form of Arithmetic Mean-Geometric Mean Inequality
$$\dfrac{ax^4+bx^{-2}}{a+b}\ge\sqrt[a+b]{x^{4a-2b}}$$
Set $4a-2b=0\iff b=2a$
Hint:
As $x^4,1/x^2>0$ using Weighted Form of Arithmetic Mean-Geometric Mean Inequality
$$\dfrac{ax^4+bx^{-2}}{a+b}\ge\sqrt[a+b]{x^{4a-2b}}$$
Set $4a-2b=0\iff b=2a$