If $a$ and $b$ are positive numbers such that $a > b$,then the minimum value of $a \sec \theta - b \tan \theta$ for $0 < \theta < \frac{\pi}{2}$ is

An open metallic tank is to be constructed with a square base and vertical sides,having a volume of $500 \,m^3$. The dimensions of the tank for the minimum area of the sheet metal used in its construction are:

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \frac{6x+\sin x}{2x+\sin x} & \text{if } x \neq 0 \\ \frac{7}{3} & \text{if } x=0 \end{cases}$. Then which of the following statements is (are) True?
$(A)$ The point $x=0$ is a point of local maxima of $f$
$(B)$ The point $x=0$ is a point of local minima of $f$
$(C)$ Number of points of local maxima of $f$ in the interval $[\pi, 6\pi]$ is $3$
$(D)$ Number of points of local minima of $f$ in the interval $[2\pi, 4\pi]$ is $1$

The maximum area of a rectangle that can be inscribed in a circle of radius $2 \text{ units}$ is

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

If $f(x) = ax + \frac{b}{x}$ where $a, b, x > 0$,then the value of $x$ for which $f(x)$ is minimum is: