Applications of the derivative
Until now, to find out whether a function was rising or falling, or whether a firm would make more or less profit when changing its scale, you had to look at the graph or try out values. With the derivative this is no longer necessary: its sign tells you, point by point, whether $f$ is increasing or decreasing, and the second derivative adds the shape of the curve. This section brings both tools together and applies them to the most classic economic problem: finding a firm's maximum profit.
Monotonicity: the sign of the derivative
An ice cream parlour has determined that its monthly profits (in hundreds of euros) depend on the number of employees hired $x$ according to the function…
An ice cream parlour has determined that its monthly profits (in hundreds of euros) depend on the number of employees hired $x$ according to the function $B(x) = -x^2 + 10x - 16$. Determine whether the profit is increasing or decreasing when the parlour has 3 employees.
Study the intervals of increase and decrease of the quadratic function $f(x) = x^2 - 6x + 8$ by analyzing the sign of its first derivative.
Study the intervals of increase and decrease of the quadratic function $f(x) = x^2 - 6x + 8$ by analyzing the sign of its first derivative.
Stationary points and relative extrema
Determine the points on the graph of the function $f(x) = x^3 - 12x + 2$ at which the tangent line is horizontal.
Determine the points on the graph of the function $f(x) = x^3 - 12x + 2$ at which the tangent line is horizontal.
Given the polynomial function $f(x) = x^3 - 6x^2 + 9x + 2$, perform a monotonicity study. To do this, determine the intervals where it is increasing and decreasing, and…
Given the polynomial function $f(x) = x^3 - 6x^2 + 9x + 2$, perform a monotonicity study. To do this, determine the intervals where it is increasing and decreasing, and find the coordinates of its local maximum and minimum points.
Curvature: concavity, convexity and inflection
Study the curvature (concavity and convexity) and find the points of inflection of the function $f(x) = 3x^4 - 4x^3$. Draw an approximate sketch of the shape of the…
Study the curvature (concavity and convexity) and find the points of inflection of the function $f(x) = 3x^4 - 4x^3$. Draw an approximate sketch of the shape of the function based solely on its second derivative.
You can do it on paper and send a photo with your answer.
Absolute extrema on a closed interval
We wish to fence a rectangular enclosure intended for a playground in a housing development. The enclosure must have an area of $600 \text{ m}^2$. One of the sides of…
We wish to fence a rectangular enclosure intended for a playground in a housing development. The enclosure must have an area of $600 \text{ m}^2$. One of the sides of the rectangle adjoins a building and does not need fencing. The price of the fence for the two lateral sides is $20 \text{ €/m}$, while the side parallel to the building uses a reinforced fence costing $60 \text{ €/m}$.
Determine the dimensions of the enclosure that minimise the total cost of the fence and calculate this minimum cost.
Maximum profit: an economic application
Everything above comes together in the favourite question of the social sciences exams: how many units should be sold to make the maximum profit?
A company that sells personalized phone cases has determined that its daily profit, in euros, follows the function $B(x) = -x^2 + 40x - 300$, where $x$ is the number of…
A company that sells personalized phone cases has determined that its daily profit, in euros, follows the function $B(x) = -x^2 + 40x - 300$, where $x$ is the number of cases sold.
- a) How many cases must they sell for the profit to be maximum?
- b) What is the amount of that maximum profit?
- c) Between which numbers of cases sold does the company make a positive profit (that is, $B(x) > 0$)?