Optimisation problems

Split the number $24$ into the sum of two positive numbers such that their product is maximum. Set up the function to be optimised and solve it using differentiation.

Which positive number exceeds its square by the greatest possible amount? Express this amount as a function of the number and maximise it using differentiation.

Find two positive numbers that sum to $18$ and whose sum of squares is minimum.

Among all rectangles with a perimeter of $36\ \text{m}$, find the dimensions of the one with maximum area and calculate that area.

A gardener wants to fence off a rectangular vegetable patch against a wall of his house, so that he only needs to fence three sides. He has $60\ \text{m}$ of fencing available.

1. a) Express the area of the vegetable patch as a function of the side $x$ perpendicular to the wall.

1. b) Find the dimensions that maximise the area and the value of that area.

A farmer has $240\ \text{m}$ of fencing to build a rectangular pen divided into two plots by an internal fence parallel to one of the sides, as shown in the figure.

1. a) Express the total area of the pen as a function of one of its sides.

1. b) Determine the dimensions that maximise the total area and calculate that area.

The area of a rectangular pen built against a wall using $40\ \text{m}$ of fencing for the other three sides is given, as a function of one of its sides $x$, by $A(x) = 40x - 2x^2$. Find the value of $x$ that maximises the area and the value of this maximum area.

Find the point on the parabola $y = x^2$ closest to the point $P(0, 3)$.

1. a) Express the square of the distance from a general point on the parabola to $P$ (minimising the square of the distance is equivalent to minimising the distance).

1. b) Find the points on the parabola that minimise this distance and calculate the minimum distance.

A clothes shop sells $80$ T-shirts each week for $\text{€}\ 12$ each. It estimates that for every euro it reduces the price, it will sell $10$ more T-shirts per week.

1. a) Express the weekly revenue $I(x)$ as a function of the price reduction $x$ in euros.

1. b) Find the price that maximises the revenue and the maximum weekly revenue.

An open-topped box is to be made from a square piece of cardboard with sides of $24\ \text{cm}$ by cutting out four equal squares of side length $x$ from the corners and folding up the sides, as shown in the figure.

1. a) Express the volume of the box $V(x)$ as a function of $x$ and state the valid domain for $x$.

1. b) Find the value of $x$ that maximises the volume.

1. c) Calculate the maximum volume and check, using the sign of $V'(x)$ on both sides, that it is indeed a maximum.

A canning company designs cylindrical tins with lids having a capacity of $500\ \text{cm}^3$, like the one in the figure.

1. a) Express the total surface area of sheet metal $S(r)$ as a function of the radius $r$ of the base, using the volume condition to express height in terms of radius.

1. b) Find the radius that minimises the amount of sheet metal (giving its exact value and an approximation rounded to two decimal places is sufficient).

1. c) What is the relationship between height and radius in the optimal tin?

Squares of side $x$ are cut out from the corners of a rectangular sheet of $60\ \text{cm} \times 30\ \text{cm}$ to construct an open box.

1. a) Express the volume $V(x)$ and state the valid domain of $x$.

1. b) Find the values of $x$ that make $V'(x)$ zero and discard the one that does not belong to the domain.

1. c) Verify that the valid value gives the maximum volume and calculate it.

Reason whether the following statement is true or false: "The maximum value of a continuous function on a closed interval is always reached at a point where $f'(x) = 0$." If false, give a counterexample using a simple function on a specific interval.

Prove, using calculus, the following general result: of all rectangles with perimeter $4p$, the one with maximum area is the square of side $p$. Set up the area function in terms of one of the sides, justify that the critical value is a maximum, and conclude.