Find two numbers whose sum is $24$ and whose product is as large as possible.

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then:

Let $f(x)=\frac{6 x^2-18 x+21}{6 x^2-18 x+17}$. If $m$ is the maximum value of $f(x)$ and $f(x) > n$ for all $x \in R$, then $14 m-7 n =$

If $f(x) = 1 + 2 \sin x + 3 \cos^2 x$ for $0 < x < 2\pi / 3$,then:

$A$ window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10 \, m$. Find the dimensions of the window to admit maximum light through the whole opening.

Let $f(x) = x^3 - 6x^2 + 12x - 3$. Then at $x = 2$,$f(x)$ has: