Monotonicity and extrema of a function
A company produces more batches and, at first, makes more money. But if it produces too many, costs shoot up and profit starts to fall. There is an exact point at which profit stops increasing and starts decreasing: a maximum. Before you can find it precisely, you need a way of measuring how a function changes: the average rate of change.
The average rate of change
To find out how $f$ changes between two points $a$ and $b$, compare how much the function rises or falls with how far you have moved along the $X$ axis.
Calculate the Average Rate of Change (ARC) of the function $f(x) = x^2 - 4x + 3$ on the interval $[2, 5]$. From the result, can you deduce whether the function is, on…
Calculate the Average Rate of Change (ARC) of the function $f(x) = x^2 - 4x + 3$ on the interval $[2, 5]$. From the result, can you deduce whether the function is, on average, increasing or decreasing on that interval?
Calculate the Average Rate of Change of the rational function $f(x) = \frac{4}{x}$ on the interval $[1, 4]$. What does the sign of the result tell us about the behaviour…
Calculate the Average Rate of Change of the rational function $f(x) = \frac{4}{x}$ on the interval $[1, 4]$. What does the sign of the result tell us about the behaviour of the function on that interval?
Increasing and decreasing
The sign of the ARC tells you whether the function goes up or down on average, but to talk about monotonicity rigorously we need a definition that does not depend on "averages": comparing the function at any pair of points in the interval.
Using the algebraic definition of monotonicity (evaluating $f(x)$ and $f(y)$ for two arbitrary values $x < y$), prove that the linear function $f(x) = 5x - 3$ is…
Using the algebraic definition of monotonicity (evaluating $f(x)$ and $f(y)$ for two arbitrary values $x < y$), prove that the linear function $f(x) = 5x - 3$ is strictly increasing throughout its domain $\mathbb{R}$.
Prove algebraically that the function $f(x) = 2x^3 + 5$ is strictly increasing for all $x \in \mathbb{R}$. To do so, take any two values $x_1$ and $x_2$ such that…
Prove algebraically that the function $f(x) = 2x^3 + 5$ is strictly increasing for all $x \in \mathbb{R}$. To do so, take any two values $x_1$ and $x_2$ such that $x_1 < x_2$ and check that $f(x_1) < f(x_2)$.
Determine the intervals of monotonicity (increasing and decreasing) of the rational function $f(x) = \frac{-3}{x-2}$. You can justify your answer by analysing the…
Determine the intervals of monotonicity (increasing and decreasing) of the rational function $f(x) = \frac{-3}{x-2}$. You can justify your answer by analysing the average rate of change in generic intervals or using the graphical properties of hyperbolas.
Maxima and minima of a function
At a maximum or a minimum, the function changes direction: it stops increasing and starts decreasing, or the other way round.
Move the sliders and see how the vertex changes:
Find analytically the vertex of the quadratic function $f(x) = 3x^2 - 12x + 7$. State whether this point corresponds to an absolute maximum or an absolute minimum of the…
Find analytically the vertex of the quadratic function $f(x) = 3x^2 - 12x + 7$. State whether this point corresponds to an absolute maximum or an absolute minimum of the function.
Find the value of the parameter $k$ so that the quadratic function $f(x) = -x^2 + kx + 5$ reaches its maximum value precisely at the x-coordinate $x = 4$. Once $k$ is…
Find the value of the parameter $k$ so that the quadratic function $f(x) = -x^2 + kx + 5$ reaches its maximum value precisely at the x-coordinate $x = 4$. Once $k$ is found, calculate this maximum value.
The profit of an electronic components company, expressed in thousands of euros, is given by the function $B(x) = -2x^2 + 80x - 300$, where $x$ is the number of batches…
The profit of an electronic components company, expressed in thousands of euros, is given by the function $B(x) = -2x^2 + 80x - 300$, where $x$ is the number of batches produced. How many batches should the company produce to maximise its profit? How much is this maximum profit?
Bounded functions
Some functions never exceed a certain value, however far $x$ advances: they are bounded.
Check whether the point $x = 3$ is a local minimum for the function $f(x) = x^2 - 6x + 10$. To do this, evaluate the function at the point itself and at two values very…
Check whether the point $x = 3$ is a local minimum for the function $f(x) = x^2 - 6x + 10$. To do this, evaluate the function at the point itself and at two values very close to its left and right, for instance $x = 2,9$ and $x = 3,1$.
Show that the function $f(x) = \frac{8}{x^2+4}$ has an absolute maximum at $x=0$. To achieve this, prove algebraically that the inequality $f(x) \le f(0)$ holds for any…
Show that the function $f(x) = \frac{8}{x^2+4}$ has an absolute maximum at $x=0$. To achieve this, prove algebraically that the inequality $f(x) \le f(0)$ holds for any real number $x$.
Prove that the quadratic function $f(x) = -3x^2 + 18x - 15$ is bounded above. Calculate its absolute maximum and determine exactly the interval in which the function is…
Prove that the quadratic function $f(x) = -3x^2 + 18x - 15$ is bounded above. Calculate its absolute maximum and determine exactly the interval in which the function is strictly decreasing.
Back to the beginning: If the relationship between production and profit follows a quadratic curve $B(x) = ax^2 + bx + c$, the optimal number of batches for maximum profit is found by calculating the $x$-coordinate of its vertex: $x_v = -\dfrac{b}{2a}$.