The function $f(x) = ax + \frac{b}{x}$ where $a, b, x > 0$ takes on the least value at $x$ equal to:

The minimum value of the polynomial $x(x + 1)(x + 2)(x + 3)$ is

The function $f(x) = 2x + 3x^{\frac{2}{3}}, x \in R$,has

${a_1, a_2, ....., a_n, .....}$ is a progression where $a_n = \frac{n^2}{n^3 + 200}$. The largest term of this progression is

Let $a \in R$ and let $f: R \rightarrow R$ be given by $f(x)=x^5-5x+a$. Then
$(A)$ $f(x)$ has three real roots if $a > 4$
$(B)$ $f(x)$ has only one real root if $a > 4$
$(C)$ $f(x)$ has three real roots if $a < -4$
$(D)$ $f(x)$ has three real roots if $-4 < a < 4$

Let the function $f: (0, \pi) \rightarrow R$ be defined by $f(\theta) = (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^4$. Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{\lambda_1 \pi, \dots, \lambda_r \pi\}$,where $0 < \lambda_1 < \dots < \lambda_r < 1$. Then the value of $\lambda_1 + \dots + \lambda_r$ is: