The maximum value of $f(x) = x^2 \log x$ on the interval $[1, e]$ is:

The function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$ where $a > 0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2 = q$,then $a =$

Find two positive numbers $x$ and $y$ such that $x+y=60$ and $x y^{3}$ is maximum.

Let $f(x) = x^2 + \frac{1}{x^2}$ and $g(x) = x - \frac{1}{x}$,$x \in R - \{-1, 1, 0\}$. If $h(x) = \frac{f(x)}{g(x)}$,then the local minimum value of $h(x)$ is:

Let the set of all positive values of $\lambda$,for which the point of local minimum of the function $f(x) = 1 + x(\lambda^2 - x^2)$ satisfies $\frac{x^2+x+2}{x^2+5x+6} < 0$,be $(\alpha, \beta)$. Then $\alpha^2 + \beta^2$ is equal to:

The minimum distance of a point on the curve $y=x^2-4$ from the origin is