The triangle of maximum area that can be inscribed in a given circle of radius $r$ is ...... .

Let $\exp(x)$ denote the exponential function $e^x$. If $f(x) = \exp\left(x^{\frac{1}{x}}\right)$ for $x > 0$,then the minimum value of $f$ in the interval $[2, 5]$ is:

$A$ wire of length $2$ units is cut into two parts,which are bent respectively to form a square of side $x$ units and a circle of radius $r$ units. If the sum of the areas of the square and the circle so formed is minimum,then:

Let $\sqrt{3}$ be the radius and $\frac{\pi}{3}$ be the semi-vertical angle of a given cone. Then the height of the right circular cylinder of maximum volume that can be inscribed in the given cone is

The minimum value of $\frac{(5 + x)(2 + x)}{1 + x}$ for non-negative real $x$ is

Let $f:[0,1] \rightarrow \mathbb{R}$ be a function. Suppose $f$ is twice differentiable,$f(0)=f(1)=0$ and satisfies $f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^x$ for $x \in[0,1]$.
$1.$ Which of the following is true for $0 < x < 1$?
$(A)$ $0 < f(x) < \infty$
$(B)$ $-\frac{1}{2} < f(x) < \frac{1}{2}$
$(C)$ $-\frac{1}{4} < f(x) < 1$
$(D)$ $-\infty < f(x) < 0$
$2.$ If the function $g(x) = e^{-x} f(x)$ assumes its minimum in the interval $[0,1]$ at $x=\frac{1}{4}$,which of the following is true?
$(A)$ $f^{\prime}(x) < f(x)$ for $x \in (0, 1/4)$
$(B)$ $f^{\prime}(x) > f(x)$ for $x \in (0, 1/4)$
$(C)$ $f^{\prime}(x) < f(x)$ for $x \in (1/4, 1)$
$(D)$ $f^{\prime}(x) > f(x)$ for $x \in (1/4, 1)$