If $$g(x)=\frac{1}{6} f\left(3 x^2-1\right)+\frac{1}{2} f\left(1-x^2\right), \forall x \in R$$, where $$f^{\prime \prime}(x) > 0, \forall x \in R$$. Then, $$g(x)$$ is increasing in the interval

If the function $$f(x)=2 x^3-9 a x^2+12 a^2 x+1$$ attains its maximum and minimum at $$p$$ and $$q$$ respectively, such that $$p^2=q$$, then $$a$$ equals

If $$y=4 x-6$$ is a tangent to the curve $$y^2=a x^4+b$$ at $$(3,6)$$, then the values of $$a$$ and $$b$$ are

Find the positive value of $$a$$ for which the equality $$2 \alpha+\beta=8$$ holds, where $$\alpha$$ and $$\beta$$ are the points of maximum and minimum, respectively, of the function $$f(x)=2 x^3-9 a x^2+12 a^2 x+1$$.