Growth and extrema using the derivative
Until now, to find where a function increases or decreases you had to look at its graph. With the derivative there is no need to draw anything: the sign of $f'(x)$ tells you, point by point, whether $f$ is going up or down, and the points where $f'$ equals zero are the candidates for a maximum or minimum. This procedure is the basis of everything that follows: sketching graphs and solving optimisation problems.
The sign of the derivative determines monotonicity
Determine the intervals where the function $f(x) = x^3 - 3x$ is increasing and decreasing by studying the sign of its derivative.
Determine the intervals where the function $f(x) = x^3 - 3x$ is increasing and decreasing by studying the sign of its derivative.
Determine the intervals of monotonicity of the function $f(x) = 2x^3 - 9x^2 + 12x - 3$.
Determine the intervals of monotonicity of the function $f(x) = 2x^3 - 9x^2 + 12x - 3$.
The figure shows the graph of the derivative $f'$ of a function $f$, which intersects the x-axis at $x = -1$ and at $x = 3$. By observing the sign of $f'$ in the figure,…
The figure shows the graph of the derivative $f'$ of a function $f$, which intersects the x-axis at $x = -1$ and at $x = 3$. By observing the sign of $f'$ in the figure, state the intervals where $f$ is increasing and decreasing and at which points it has local extrema.
Stationary points: where to look for extrema
To classify a stationary point $x_0$ there are two equivalent methods:
Find the local extremum of the function $f(x) = x^2 - 6x + 5$ using the derivative, and determine with reasons whether it is a maximum or a minimum.
Find the local extremum of the function $f(x) = x^2 - 6x + 5$ using the derivative, and determine with reasons whether it is a maximum or a minimum.
Consider the function $f(x) = x^3 - 3x^2 - 9x + 2$. a) Calculate its derivative and find its stationary points. b) Determine the intervals where the function is…
Consider the function $f(x) = x^3 - 3x^2 - 9x + 2$.
- a) Calculate its derivative and find its stationary points.
- b) Determine the intervals where the function is increasing and decreasing.
- c) Classify each stationary point as a local maximum or local minimum and state the value of the function at these points.
The function $f(x) = x^3 + ax^2 + bx$ has a local extremum at the point $(1, -2)$. a) Set up the two conditions that $a$ and $b$ must satisfy and calculate their values.…
The function $f(x) = x^3 + ax^2 + bx$ has a local extremum at the point $(1, -2)$.
- a) Set up the two conditions that $a$ and $b$ must satisfy and calculate their values.
- b) Using these values, locate the other local extremum of the function and classify both.
Absolute extrema on a closed interval
Calculate the stationary points of the function $f(x) = x^3 - 12x + 1$, that is, the points where its derivative is zero.
Calculate the stationary points of the function $f(x) = x^3 - 12x + 1$, that is, the points where its derivative is zero.
A small workshop manufactures $x$ bicycles per month, and the average cost per bicycle, in euros, is given by $C(x) = x + \frac{1600}{x}$, with $x > 0$. Find the number…
A small workshop manufactures $x$ bicycles per month, and the average cost per bicycle, in euros, is given by $C(x) = x + \frac{1600}{x}$, with $x > 0$. Find the number of bicycles that minimises the average cost and the value of this minimum cost, confirming with the sign of the derivative that it is indeed a minimum.
Let $f(x) = x^3 - 3x + 1$ be defined on the closed interval $[-2, 3]$. a) Locate its stationary points within the interval. b) Calculate the value of $f$ at the…
Let $f(x) = x^3 - 3x + 1$ be defined on the closed interval $[-2, 3]$.
- a) Locate its stationary points within the interval.
- b) Calculate the value of $f$ at the stationary points and at the endpoints of the interval.
- c) Determine the absolute maximum and minimum of $f$ on $[-2, 3]$ and explain why it is not sufficient to check only the stationary points.