Optimisation 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?

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?

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 cinema with $400$ seats sells tickets for $8\ \text{ €}$ and fills every session. The manager estimates that for every $0,50\ \text{ €}$ increase in price, they will lose $20$ viewers.

1. a) If $x$ is the number of $0,50\ \text{ €}$ 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 $2\ \text{ €}$ so as not to empty the auditorium. Is it worth raising it by those $2\ \text{ €}$? Justify your answer using the model.

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

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 minimises 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.

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.

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.