Monotonicity and Extrema of a Function
Marcus prints custom T-shirts for school teams. His profit is $P(x) = -2x^2 + 40x - 30$ dollars when he prints $x$ dozen shirts. At first, printing more shirts means more profit. Past a certain point, rush fees and wasted supplies push the profit back down. Somewhere there is a best production level, a maximum. To find it with confidence, you first need ways to describe how a function changes.
Average Rate of Change
To describe how $f$ changes between two inputs $a$ and $b$, compare how far the output moved with how far the input moved.
A positive ARC means the output rises on average. A negative ARC means it falls on average. The units of the ARC are (units of $y$) per (unit of $x$), such as miles per hour or dollars per batch.
When you compare two functions on the same interval, the larger ARC is the one that grows faster on average there.
Calculate the Average Rate of Change (ARC) of the function $f(x) = x^2 - 4x + 3$ in 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 $f(x) = x^2 - 4x + 3$ in the interval $[2, 5]$. From the result, can you deduce whether the function is, on average, increasing or decreasing in that interval?
Calculate the Average Rate of Change of the rational function $f(x) = \frac{4}{x}$ in the interval $[1, 4]$. What does the sign of the result indicate about the behavior…
Calculate the Average Rate of Change of the rational function $f(x) = \frac{4}{x}$ in the interval $[1, 4]$. What does the sign of the result indicate about the behavior of the function in that interval?
In a laboratory culture, the number of bacteria $N(t)$ as a function of time $t$ (in hours) follows the law $N(t) = 20t^2 + 100t + 500$. Calculate the Average Rate of…
In a laboratory culture, the number of bacteria $N(t)$ as a function of time $t$ (in hours) follows the law $N(t) = 20t^2 + 100t + 500$. Calculate the Average Rate of Change in the interval $[1, 4]$ and interpret the meaning of the result in the context of the problem.
Increasing and Decreasing Functions
The ARC describes an average. To say that a function is increasing, you need a statement that holds for every pair of points in the interval.
In words: moving right always moves the graph up (increasing) or always down (decreasing). The usual strategy for a proof is to take any $x1 < x_2$, form $f(x_2) - f(x_1)$, factor it, and decide its sign.
Hyperbolas. For $f(x) = \dfrac{5}{x}$ and $0 < x_1 < x_2$:
$$f(x_2) - f(x_1) = \frac{5}{x_2} - \frac{5}{x_1} = \frac{5(x_1 - x_2)}{x_1x_2}.$$The numerator is negative and the denominator is positive, so $f(x2) < f(x_1)$. The function is decreasing on $(0, \infty)$. The same argument works on $(-\infty, 0)$, because $x_1x_2 > 0$ there too.
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 over its entire domain $\mathbb{R}$.
Prove algebraically that the function $f(x) = 2x^3 + 5$ is strictly increasing for all $x \in \mathbb{R}$. To do this, 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 this, 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 analyzing 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 analyzing the Average Rate of Change in generic intervals or by relying on the graphical properties of hyperbolas.
Extrema and the Vertex of a Parabola
At a peak or a valley, a function changes direction: it stops increasing and starts decreasing, or the other way around. These are the extrema (singular: extremum).
An absolute extremum is also a local one. The converse is false.
In vertex form $f(x) = a(x - h)^2 + k$, the vertex is simply $(h, k)$.
Move the sliders and watch the vertex travel:
Checking a candidate numerically. To test whether $x_0$ is a local minimum, compare $f(x_0)$ with values just to its left and right. For $f(x) = x^2 - 4x + 7$ and $x_0 = 2$: $f(2) = 3$, $f(1.9) = 3.61 - 7.6 + 7 = 3.01$, and $f(2.1) = 4.41 - 8.4 + 7 = 3.01$. Both neighbors are higher, which is consistent with a local minimum at $x = 2$.
Finding a parameter. If a problem tells you where the vertex is, set $-\dfrac{b}{2a}$ equal to that $x$-value.
Find analytically the vertex of the quadratic function $f(x) = 3x^2 - 12x + 7$. State whether this point corresponds to an absolute maximum or minimum of the function.
Find analytically the vertex of the quadratic function $f(x) = 3x^2 - 12x + 7$. State whether this point corresponds to an absolute maximum or minimum of the function.
Check if 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…
Check if 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 example $x = 2.9$ and $x = 3.1$.
Find the value of the parameter $k$ so that the quadratic function $f(x) = -x^2 + kx + 5$ reaches its maximum value exactly 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 exactly at the x-coordinate $x = 4$. Once $k$ is found, calculate this maximum value.
Optimization with Quadratic Models
When a quantity is modeled by a quadratic, the best possible value is the vertex. Setting up the model is half of the work.
A stone is thrown vertically upwards. The height in feet it reaches as a function of time in seconds is given by the expression $h(t) = -5t^2 + 30t + 2$. Calculate the…
A stone is thrown vertically upwards. The height in feet it reaches as a function of time in seconds is given by the expression $h(t) = -5t^2 + 30t + 2$. Calculate the time at which the stone reaches its maximum height and determine what this height is.
The temperature inside a chemical reactor during the first few hours of a process follows the function $T(t) = -0.2t^2 + 2.4t + 20$, with $t$ in hours and $T$ in degrees…
The temperature inside a chemical reactor during the first few hours of a process follows the function $T(t) = -0.2t^2 + 2.4t + 20$, with $t$ in hours and $T$ in degrees Fahrenheit. Determine the maximum temperature reached and the time interval in which the temperature of the reactor is decreasing.
A farmer has $120$ feet of fencing to enclose a rectangular vegetable patch. One of the sides of the patch will be against the straight wall of a barn, so no fencing…
A farmer has $120$ feet of fencing to enclose a rectangular vegetable patch. One of the sides of the patch will be against the straight wall of a barn, so no fencing will be needed on that side. Find the dimensions of the patch that make its area maximum and calculate this area.
Back to the start: Marcus's best choice is 10 dozen shirts, for a profit of $\$170$. That is the vertex $x = -\dfrac{b}{2a}$ of his profit parabola.
Reading Extrema and Monotonicity from a Graph
On a graph, read left to right.
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Where the graph rises, $f$ is increasing. Where it falls, $f$ is decreasing.
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A peak where the graph switches from rising to falling is a local maximum. A valley where it switches from falling to rising is a local minimum.
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Report the intervals using $x$-values and the extrema as points $(x, y)$.
Here $A$ is a local maximum and $B$ is a local minimum. The function is increasing on $(-\infty, x_1)$ and $(x_2, \infty)$, and decreasing on $(x_1, x_2)$. Writing open intervals is the standard convention, since at the turning points the function stops one behavior and starts the other.
Absolute extrema from a graph. Ask where the graph is highest or lowest over the whole domain. If the graph keeps going up forever or down forever, there is no absolute maximum or minimum on that side. The curve above has neither, because it rises to the right and falls to the left without end.
Piecewise graphs. Watch the dots. A closed dot is a point of the graph, and an open dot is not. An absolute minimum can sit at a closed dot at the end of a piece, and an open dot can never be an extremum, because the function never takes that value.
Observe the following graph of a polynomial function. Identify the coordinates of the points $A$ and $B$, and classify them as relative maxima or minima. Then, write the…
Observe the following graph of a polynomial function. Identify the coordinates of the points $A$ and $B$, and classify them as relative maxima or minima. Then, write the intervals where the function is strictly increasing and where it is strictly decreasing.
The following graph represents a quadratic function. Determine the coordinates of its vertex $V$. Is it a maximum or a minimum? Is it relative, absolute, or both?
The following graph represents a quadratic function. Determine the coordinates of its vertex $V$. Is it a maximum or a minimum? Is it relative, absolute, or both?
The following graph shows a piecewise function. From it, determine the intervals where the function is increasing and decreasing. Also identify the coordinates of the…
The following graph shows a piecewise function. From it, determine the intervals where the function is increasing and decreasing. Also identify the coordinates of the absolute minimum and state if there is any maximum. Each grid square represents $1$ unit on both axes.
Piecewise and Rational Functions
Piecewise functions
Analyze each piece on its own domain, then examine the point where the rule changes.
When the junction is an extremum, it is because the function increases on one side and decreases on the other, or because a jump puts the point above or below its neighbors. Always compare $f$ at the junction with values just to either side.
Rational functions
You will often analyze these from a graph. Do not join intervals across a vertical asymptote, because the function is undefined there.
For $f(x) = x + \dfrac{1}{x}$: it is increasing on $(-\infty, -1)$ and $(1, \infty)$ and decreasing on $(-1, 0)$ and $(0, 1)$. The local maximum $(-1, -2)$ is lower than the local minimum $(1, 2)$. That is possible because the two branches are separated by the asymptote.
Confirming with ARC. If the ARC is positive on one interval and negative on the next, the function must change direction somewhere between them. For the same $f$, the ARC on $[0.5, 1]$ is $\dfrac{2 - 2.5}{0.5} = -1$ and the ARC on $[1, 3]$ is $\dfrac{3.33 - 2}{2} \approx 0.67$. The sign change signals a minimum near $x = 1$. This is evidence, not proof, since averages can hide changes.
Given the piecewise function: $f(x) = \begin{cases} -x^2 + 6x - 5 & \text{if } x \le 4 \\ 2x - 5 & \text{if } x > 4 \end{cases}$ Analyze analytically its intervals of…
Given the piecewise function:
$f(x) = \begin{cases} -x^2 + 6x - 5 & \text{if } x \le 4 \\ 2x - 5 & \text{if } x > 4 \end{cases}$
Analyze analytically its intervals of monotonicity and find the coordinates of its relative or absolute extrema. Justify whether the point where the branch changes ($x=4$) represents any type of extremum.
The following graph represents a rational function with a vertical and an oblique asymptote. Observing the graph, identify the approximate coordinates of its relative…
The following graph represents a rational function with a vertical and an oblique asymptote. Observing the graph, identify the approximate coordinates of its relative extrema (local maximum and minimum). Then, write the intervals in which the function is increasing and in which it is decreasing.
The concentration of a medicine in the blood, measured in mg/L, as a function of the hours elapsed since its administration, is modeled by the function…
The concentration of a medicine in the blood, measured in mg/L, as a function of the hours elapsed since its administration, is modeled by the function $C(t) = \frac{12t}{t^2+4}$ for $t \ge 0$. Observe the attached graph. Estimate the time at which the concentration is maximum. Then, calculate the Average Rate of Change in the intervals $[0, 2]$ and $[2, 6]$ to confirm analytically the change in monotonicity.
Bounded Functions
Some functions never rise above a certain height, no matter how far $x$ goes. They are bounded.
Prove 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 $f(x) \le f(0)$ is satisfied…
Prove 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 $f(x) \le f(0)$ is satisfied 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.