Find the maximum value of \(A(x)=12x-x^2\) for \(0\le x\le12\).

A rectangle has perimeter \(40\) cm. Find the side lengths that maximise its area.

Find the minimum value of \(f(x)=x^2+6x+11\) on the real line.

A right triangle has legs \(x\) cm and \(y\) cm with \(x+y=18\). Find the maximum possible area.

Find the point on the line \(y=2x+1\) closest to the origin.

An open-top box is made by cutting squares of side length \(x\) cm from the corners of a \(20\) cm by \(12\) cm sheet and folding up the sides. Find the value of \(x\) that maximises volume, with \(0<x<6\).

Find the positive number \(x\) that minimises \(C(x)=x+\frac{16}{x}\).

A cylindrical can has fixed volume \(500\pi\) cm\(^3\). If the radius is \(r\) cm and height is \(h\) cm, find the radius that minimises surface area \(S=2\pi r^2+2\pi rh\).

Find the maximum value of \(f(x)=x(10-x)^2\) for \(0\le x\le10\).

A farmer has \(60\) m of fencing for three sides of a rectangular pen, with the fourth side along a wall. Find the dimensions that maximise the enclosed area.