Growth and extrema using the derivative
Suppose you are cycling from Galway out to the Cliffs of Moher and want to know where the road climbs and where it descends, without looking at the whole route. 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
Back to the cycling example: on a road, the slope tells you whether you are climbing or descending, and a stationary point is where the road is momentarily level, at the top of a hill or the bottom of a dip.
Strictly increasing everywhere: a condition on a parameter
If $f$ is increasing on all of $\mathbb{R}$, then $f'(x) \geq 0$ for all $x$. When $f'$ is a quadratic with positive leading coefficient, its graph must never go below the x-axis, so the discriminant satisfies $b^2 - 4ac \leq 0$. A single isolated zero of $f'$ (discriminant $= 0$) does not stop $f$ being strictly increasing.
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.
Consider the family of functions $f_k(x) = x^3 - 3kx^2 + 9x - 1$, where $k \in \mathbb{R}$ is a parameter. a) Calculate $f'k(x)$ and determine for which values of $k$…
Consider the family of functions $f_k(x) = x^3 - 3kx^2 + 9x - 1$, where $k \in \mathbb{R}$ is a parameter.
- a) Calculate $f'_k(x)$ and determine for which values of $k$ the function is strictly increasing on all of $\mathbb{R}$. Explain what happens to the graph and the tangent line in the particular case $k = \sqrt{3}$.
- b) Find the values of $k$ for which the function has exactly two distinct stationary points. Classify them as a relative maximum or minimum, justifying your answer according to the signs adopted by the derivative along the real line.
- c) For the specific case $k = 2$, restrict the function to the closed interval $[0, 5]$. Determine with reasoning all candidates for absolute extrema, evaluate the function at them, and locate the absolute maximum and minimum on that interval. In light of the result, argue why on a closed interval the absolute maximum does not necessarily have to coincide with the relative maximum.
Counting solutions of an equation
If $f$ is strictly increasing and continuous, its graph crosses the x-axis at most once. If $f$ changes sign between two values, the graph must cross the axis in between. Together these give exactly one solution.
Stationary points: where to look for extrema
To classify a stationary point $x_0$ there are two equivalent methods:
Finding a parameter from an extremum
If you are told a function has a local extremum at $x=x_0$, then $f'(x_0)=0$. If you are also told the point lies on the curve, then $f(x_0)$ equals the given value. Each condition gives one equation in the unknown parameters.
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.
Find the value of the parameter $b$ such that the function $f(x) = x^2 + bx + 3$ has a local minimum at the point $x = 2$. With this value of $b$, calculate the minimum…
Find the value of the parameter $b$ such that the function $f(x) = x^2 + bx + 3$ has a local minimum at the point $x = 2$. With this value of $b$, calculate the minimum value of the function.
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
Behaviour as $x \to \pm\infty$
To describe a function on all of $\mathbb{R}$, also ask what happens to $f(x)$ as $x$ becomes very large. Divide by the highest power of $x$, or use the fact that $e^{-x} \to 0$ as $x \to +\infty$ and that the exponential beats any power of $x$. Combined with the stationary points, this tells you whether a local extremum is also the absolute one.
For functions with $e^{kx}$, recall that $\frac{d}{dx}e^{kx}=ke^{kx}$, and use the product rule $(uv)'=u'v+uv'$.
For a fraction, use the quotient rule (it is in the Formulae and Tables booklet): $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$. Since $v^2>0$, the sign of $f'$ is the sign of the numerator $u'v-uv'$.
For a term like $\frac{k}{x}$, first write it as a power: $\frac{k}{x}=kx^{-1}$, so $\frac{d}{dx}\left(\frac{k}{x}\right)=-kx^{-2}=-\frac{k}{x^2}$.
Fully analyse the monotonicity of the function $f(x) = \frac{x}{x^2 + 1}$: increasing and decreasing intervals, local extrema with their values, and behaviour as…
Fully analyse the monotonicity of the function $f(x) = \frac{x}{x^2 + 1}$: increasing and decreasing intervals, local extrema with their values, and behaviour as $x \to \pm \infty$.
Study the monotonicity and extrema of the function $f(x) = x \times e^{-x}$ on all of $\mathbb{R}$, indicating the value of the local extremum and the limit of the…
Study the monotonicity and extrema of the function $f(x) = x \times e^{-x}$ on all of $\mathbb{R}$, indicating the value of the local extremum and the limit of the function as $x \to +\infty$.
Given the polynomial function $f(x) = 2x^3 - 9x^2 + 12x$ restricted to the closed interval $[0, 3]$: a) Before calculating any derivative, evaluate mentally or quickly…
Given the polynomial function $f(x) = 2x^3 - 9x^2 + 12x$ restricted to the closed interval $[0, 3]$:
- a) Before calculating any derivative, evaluate mentally or quickly the endpoints of the interval, $f(0)$ and $f(3)$, as well as the value at $f(1)$. From these three data points, make a reasoned estimate of whether you think the absolute maximum will be reached in the interior of the interval or at the right endpoint $x = 3$.
- b) Calculate the derivative function $f'(x)$, determine the stationary points that belong to the interval $[0, 3]$, and classify them by studying the sign of $f'$.
- c) Find the exact absolute maximum and minimum on $[0, 3]$ by evaluating the relevant candidates (interior stationary points and interval endpoints) and check whether your initial estimate was correct.
Met Éireann records the temperature evolution during a summer day at a station in the Irish midlands (Birr, Co. Offaly). The temperature $T$ (in degrees Celsius)…
Met Éireann records the temperature evolution during a summer day at a station in the Irish midlands (Birr, Co. Offaly). The temperature $T$ (in degrees Celsius) throughout the $24$ hours of the day is modeled by the function:
$$T(t) = -0.01t^3 + 0.3t^2 - 1.92t + 32, \quad t \in [0, 24]$$where $t$ represents the time elapsed in hours since midnight ($t = 0$).
- a) Calculate the derivative $T'(t)$ and determine the intervals of the day during which the temperature was increasing and during which it was decreasing.
- b) Find at what exact time the relative (local) minimum temperature of the early morning was recorded and at what time the absolute maximum was reached, calculating the thermal value at both moments.
- c) Interpret the sign of the derivative $T'(t)$ in the period from $14:00$ to $16:00$ hours to explain why the daily maximum temperature does not coincide with solar noon (the moment of maximum insolation), but instead exhibits thermal lag.
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.