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?

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.

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?

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.

The profit of a factory, in thousands of euros, when producing $x$ tonnes per week is $B(x) = -2x^{2} + 80x - 300$.

1. a) What weekly production level maximises profit?

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

A company has a revenue $I(q) = -0.2q^{2} + 60q$ and costs $C(q) = 20q + 800$, in euros, 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 maximises 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 market study estimates that the weekly profit of a shop, in euros, 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 shop 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 $B$.

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.

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.