The minimum value of $4e^{2x} + 9e^{-2x}$ is:

The difference between the greatest and the least values of the function $f(x) = x(\ln x - 2)$ on the interval $[1, e^2]$ is:

Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. Then which of the following statements are true?
$P: x=0$ is a point of local minima of $f$
$Q: x=\sqrt{2}$ is a point of inflection of $f$
$R: f^{\prime}$ is increasing for $x>\sqrt{2}$

$A$ rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$,the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$

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:

If $f(x) = \int\limits_0^x {{e^{\frac{{ - {t^2}}}{2}}}} \left( {1 - {t^2}} \right)\,dt$,then $f(x)$ is minimum at $x = \dots$