The minimum value of $2x + 3y,$ when $xy = 6,$ is

The function $f(x) = x^3 - 6x^2 + 9x + 2$ has a maximum value when $x$ is

$A$ rectangle has one side on the positive $y-$ axis and one side on the positive $x-$ axis. The upper right hand vertex lies on the curve $y = \frac{\ln x}{x^2}$. The maximum area of the rectangle is

If $g(x) = 2f(2x^3 - 3x^2) + f(6x^2 - 4x^3 - 3)$,$\forall x \in R$ and $f''(x) > 0$,$\forall x \in R$,then $g'(x) > 0$ for $x$ belonging to

If the minimum value of $f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}$ for $x > 0$ is $14$,then the value of $\alpha$ is equal to:

Let $f(x)$ be a polynomial of degree $3$ such that $f(-1)=10$,$f(1)=-6$,$f(x)$ has a critical point at $x=-1$,and $f^{\prime}(x)$ has a critical point at $x=1$. Then $f(x)$ has a local minima at $x=$