The minimum value of the expression $7 - 20x + 11x^2$ is (in $/11$)

The minimum value of $f(a) = (2a^2 - 3) + 3(3 - a) + 4$ is

If the curved surface area of a right circular cylinder inscribed in a sphere of radius $22 \ cm$ is maximum,then the height of the cylinder will be:

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 $x \in (1/4, 1)$

Let $f: R \to R$,$f(x) = ax^3 + bx^2 + cx + d$ have no extreme value. Then which of the following is always correct?

If $ab = 2a + 3b$ where $a > 0$ and $b > 0$,then the minimum value of $ab$ is: