Optimization and Complete Study of Functions

Let $f(x) = x^{2} - 4x + 3$.

1. a) Calculate $f'(x)$ and find the point where it equals zero.

1. b) Is this point a maximum or a minimum? Justify your answer by studying the sign of $f'$ on both sides.

The derivative of a function is $f'(x) = 2x - 6$.

1. a) At which point does $f'$ equal zero?

1. b) Where is $f$ increasing and where is it decreasing?

1. c) What type of extremum does $f$ have, and at which point?

The derivative of a function $f$ equals zero at $x = 2$ and at $x = 5$, and its sign is given by the table:

• In $(-\infty, 2)$: $f'(x) > 0$.

• In $(2, 5)$: $f'(x) < 0$.

• In $(5, +\infty)$: $f'(x) > 0$.

Classify the points $x = 2$ and $x = 5$ as relative maximum or relative minimum, justifying your answer.

Determine whether the statement is true or false: "if $f'(3) = 0$, then $f$ has a relative maximum or minimum at $x = 3$". If false, give a specific counterexample and explain it.

The figure shows the weekly profit of an ice cream shop as a function of the ice cream price: the parabola intersects the price axis at $\$5$ and at $\$25$.

1. a) Using the symmetry of the parabola, which price maximises profit?

1. b) For which prices is the profit zero? What do these values mean?

1. c) For which prices is the profit negative? Interpret this.

The profit of a factory, in thousands of pesos, when producing $x$ metric tons per week is $B(x) = -40x^{2} + 1\,600x - 6\,000$.

1. a) What weekly production level maximizes profit?

1. b) What is this maximum profit, in pesos?

A market study estimates that the weekly profit of a store, in pesos, when selling $x$ units of a product is $B(x) = -x^{2} + 140x - 3,600$.

1. a) How many units must it sell to obtain maximum profit? How much is this profit?

1. b) For which sales levels does the store lose money?

1. c) If, due to storage limitations, it cannot sell more than $50$ units, what is its best decision then? Justify your answer using the monotonicity of $B$.

There are $40$ m of fencing available to enclose a rectangular pen on all four sides.

1. a) If one side measures $x$, express the area of the pen in terms of $x$.

1. b) What dimensions maximise the area? What shape is the pen then?

We want to fence $100$ m to make a rectangular vegetable patch against a wall, so that only three sides need to be fenced, as shown in the figure.

1. a) If $x$ is the length of each side perpendicular to the wall and $y$ is the length of the parallel side, express $y$ as a function of $x$ and write the area function $A(x)$.

1. b) Calculate the dimensions that maximise the area.

1. c) What is this maximum area?

From a square sheet of cardboard with side length $24$ cm, four identical squares of side length $x$ are cut out from the corners to construct, by folding, an open-topped box, as shown in the figure.

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

1. b) Calculate $V'(x)$ and find its critical points, discarding those that do not belong to the domain.

1. c) What value of $x$ maximises the volume? What is this maximum volume?

A canning company wants to manufacture a box with a square base and a lid, with a volume of $1\ \text{dm}^{3}$ ($1,000\ \text{cm}^{3}$), using the minimum amount of cardboard.

1. a) If $x$ is the side length of the base and $y$ is the height, express $y$ in terms of $x$ using the volume condition.

1. b) Write the total surface area function $S(x)$ and calculate its derivative.

1. c) Find the dimensions that minimise the surface area. What shape is the optimal box?

Find the point on the parabola $y = x^{2}$ closest to the point $P = \left(0, \frac{3}{2}\right)$.

1. a) Express the square of the distance from a generic point $(x, x^{2})$ on the parabola to the point $P$.

1. b) Differentiate this expression and find its critical points.

1. c) Determine which points on the parabola are at a minimum distance from $P$ and calculate that minimum distance. Why are there two solution points?

A window has the shape of a rectangle surmounted by a semicircle (arched window). The total perimeter of the frame is $12$ m and it is desired to let in as much light as possible, that is, the area should be maximum.

1. a) If $r$ is the radius of the semicircle and $h$ the height of the rectangle, write the relation between $r$ and $h$ imposed by the perimeter.

1. b) Express the total area $A(r)$ as a function of $r$ only.

1. c) Find the value of $r$ that maximises the area, leaving the result in terms of $\pi$, and calculate the full dimensions of the window.

A student solves an optimisation problem as follows: "We have to divide $50$ m of string into two pieces to enclose two square areas of side $x$ and side $y$. The total area is $A = x^{2} + y^{2}$. Since $4x + 4y = 50$, it follows that $y = 12.5 - x$, so $A(x) = x^{2} + (12.5 - x)^{2}$. Differentiating: $A'(x) = 4x - 25 = 0$, hence $x = 6.25$. Therefore, the maximum area is $6.25\ \text{m}^{2}$."

1. a) Find the two errors in their conclusion (one regarding the value of $6.25$ and another regarding whether it is a maximum or a minimum).

1. b) Solve the problem correctly: how should the string be cut to minimise the total area? And to maximise it? (Think about what happens at the boundaries of the domain of $x$.)

A company has a revenue $I(q) = -4q^{2} + 1\,200q$ and costs $C(q) = 400q + 16\,000$, in pesos, where $q$ are the units produced and sold. The figure shows both curves.

1. a) Write the profit function $B(q)$ and calculate its derivative.

1. b) What production level maximizes profit? What is this maximum profit?

1. c) Calculate the break-even points (the values of $q$ for which $B(q) = 0$) and relate them to the intersection points in the figure.

A cinema with $400$ seats sells tickets for $\$160$ and fills every session. The manager estimates that for every $\$10$ increase in price, they will lose $20$ viewers.

1. a) If $x$ is the number of $\$10$ price increases, write the revenue function $I(x)$.

1. b) What ticket price maximises the revenue? How much is this maximum revenue?

1. c) Above what price would the revenue be less than the current full house revenue?

1. d) The manager decides not to raise the price by more than $\$40$ so as not to empty the auditorium. Is it worth raising it by those $\$40$? Justify your answer using the model.

The total cost of producing $q$ units of a product is $C(q) = q^{2} + 30q + 25$, in pesos.

1. a) Write the average cost per unit function, $CM(q) = \frac{C(q)}{q}$, for $q > 0$.

1. b) Calculate the value of $q$ that minimizes the average cost and verify, by studying the sign of $CM'(q)$, that it is a minimum.

1. c) Calculate the marginal cost $C'(q)$ at that same value and check that it coincides with the minimum average cost. Is it a coincidence? Investigate the relationship between $CM'(q)$ and $C'(q) - CM(q)$ and state the general conclusion.

Reasonably assess whether these statements are true or false, providing examples to justify your answer:

1. a) "Every continuous function on a closed interval $[a, b]$ achieves a maximum value and a minimum value on it."

1. b) "Every continuous function on an open interval $(a, b)$ achieves a maximum value and a minimum value on it."

1. c) "If a function has a local maximum at $x = c$, that is also its absolute maximum over its entire domain."

Find the absolute maximum and minimum values of the function $f(x) = x^{3} - 6x^{2} + 9x + 1$ on the interval $[0, 4]$.

1. a) Calculate the critical points inside the interval.

1. b) Evaluate the function at the critical points and at the endpoints of the interval.

1. c) Conclude: indicate the absolute maximum and minimum and at which points they are reached. Are you surprised by any of the results? Comment on them.

Let $f(x) = x^{3} - 3x$.

1. a) Calculate $f'(x)$.

1. b) At which points does the derivative equal zero?

1. c) State the intervals where $f$ is increasing.

Study and sketch the function $f(x) = x^{3} - 3x^{2} + 2$:

1. a) Axes intercepts.

1. b) Intervals where the function is increasing and decreasing, and local extrema.

1. c) Curvature and point of inflection.

1. d) Sketch the graph.

You can do it on paper and send a photo with your answer.

The figure shows the graph of a polynomial function $f$. Read the values you need from the grid.

1. a) State the intervals where the function is increasing and decreasing.

1. b) What is the sign of $f'(x)$ in each of those intervals? What is the value of $f'$ at the points where the function changes from increasing to decreasing or from decreasing to increasing?

1. c) What are the coordinates of the relative maximum and minimum? Are they also absolute? Justify your answer by looking at the branches of the graph.

Carry out a complete study of the function $f(x) = x^{3} - 12x$ and sketch its graph:

1. a) Domain, axis intercepts and symmetry.

1. b) Intervals where the function is increasing and decreasing, and local extrema.

1. c) Curvature and point of inflection.

1. d) Sketch the graph using the information obtained.

You can do it on paper and send a photo with your answer.

Study the function $f(x) = \frac{x}{x^{2} + 1}$:

1. a) Domain and symmetry (check that it is odd).

1. b) Horizontal asymptote (calculate the limit as $x \to +\infty$).

1. c) Relative extrema and monotonicity.

1. d) Using the above information, sketch its complete graph.

You can do it on paper and send a photo with your answer.

The figure shows the graph of a rational function with a vertical asymptote $x = 2$ and a horizontal asymptote $y = 1$.

1. a) State the domain of the function and calculate, from the graph, $\lim_{x \to 2^{-}} f(x)$, $\lim_{x \to 2^{+}} f(x)$ and $\lim_{x \to +\infty} f(x)$.

1. b) Estimate the coordinates of the local maximum marked with "?" and explain what condition the derivative must satisfy at that point.

1. c) State the intervals where each branch is increasing and decreasing, and reason why the function has no extrema on the right branch.

1. d) A classmate claims that the function could be $g(x) = 1 + \frac{3}{x - 2}$, which has the same asymptotes. Calculate $g'(x)$, study its sign and reason why that function cannot be the one in the graph.