$A$ square piece of tin of side $18 \, cm$ is to be made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible (in $, cm$)?

The function $f(x) = x e^{-x}, \forall x \in R$ attains a maximum value at $x$ equal to:

The least area of a circle circumscribing any right triangle of area $S$ is

Let $f: D \rightarrow R$, $D \subseteq R$, $c \in D$ and $r$ be a non-zero real number. Consider the following statements:
$Y$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $rf$
$M$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $r+f$
Which of the following is correct?

What is the maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$?

The set of value$(s)$ of $a$ for which the function $f(x) = \frac{a x^3}{3} + (a + 2) x^2 + (a - 1) x + 2$ possesses a negative point of inflection.