Tools, electronics, clothing, and even food are all produced in factories on a large scale. Factories often face a common problem: they need to minimize expenses while maximizing production.
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Team Maxima and Minima Teachers
Calculus is a powerful tool when modeling a wide variety of situations. The use of functions allows us to gain a better understanding of a particular scenario. How can calculus be used in our factory example?
A function is a rule that assigns an output to every given input in its domain. Some of these outputs can be greater than others, which leads to the following questions:
Is there an output that is greater than every other output?
Is there an output that is less than every other output?
These high and low output values are known as extrema. The highest output value is a maximum and the lowest output value is a minimum. In the plural, they are known as maxima and minima, respectively.
There are two types of maxima and minima: global and local. We will explore these two types of extrema and how to find them.
The absolute maximum of a function, or global maximum, is the greatest output of the function over its entire domain.
The Absolute Maximum, or Global Maximum, of a function is the greatest output in the range of the function. If is the absolute maximum of a function
, then
for all
in the domain of the function.
The absolute minimum, or global minimum, is defined similarly as the least output of the function over its entire domain.
The Absolute Minimum, or Global Minimum, of a function is the smallest output in the range of the function. If is the absolute minimum of a function
, then
for all
in the domain of the function.
Not all functions have a global maximum or a global minimum. Functions may have one, none, or both.
Parabolas are a good example of functions that have a global maximum or a global minimum. Let's take a look at the graph of the function :
Fig. 1: Graph of the parabola.
This parabola has a minimum at the vertex,. Therefore, it has a global minimum located at
, and its value is the y-value of the vertex, which is
.
The parabola is defined for all real numbers, so the outputs will continue increasing as increases or decreases. Therefore the function does not have a global maximum.
But what happens if the function is not defined over all real numbers? Let's take a look at the next example.
Consider the following graph:
Fig. 2: Graph of the parabola over a smaller domain.
This is the same function as before, , but its domain is now restricted to
. Its maximum occurs at the point
. Therefore, it has a global maximum at
, and its value is
.
It is worth noting that the parabola still has the same global minimum value of 1.
Some functions might not have either maxima or minima!
This time we will take a look at the graph of the linear function .
Fig. 3:Graph of the linear function.
This function is defined for all real numbers. Its outputs will continue decreasing to the left and increasing to the right. Therefore this function does not have a global maximum or a global minimum.
A function's relative maximum, or local maximum, is an output that is greater than the outputs directly next to it. This implies that we can find an interval around it such that this output is greater than all the other outputs of the values in the chosen interval.
A function is said to have a Relative Maximum, or Local Maximum, at
if there exists an interval
containing
such that
for all
in that interval. The value
is a relative maximum.
A relative minimum, or local minimum, is defined similarly as an output that is less than the outputs directly next to it.
A function is said to have a Relative Minimum, or Local Minimum at
if there exists an interval
containing
such that
for all
in that interval. The value
is a relative minimum.
But how do we find the local maximum or local minimum? Let's take a look at the graph of the function.
Consider the graph of a cubic function.
Fig. 4: Graph of a cubic function with relative extrema.
We can identify the relative extrema as the peaks and valleys of the graph. We can see a peak at , so a local maximum is there. We can also see a valley at
, which means that a local minimum is there.
Note that this function does not have a global minimum because its values continue decreasing to the left. Likewise, it does not have a global maximum because its values continue increasing to the right.
It is also worth noting that the function switches from increasing (positive slope) to decreasing (negative slope) at a local maximum. Likewise, at a local minimum, the function switches from decreasing to increasing. At these points, if the graph is a smooth curve, the slope of the function is equal to 0. This is an important observation because it will allow us to use calculus, in particular derivatives, in finding relative extrema when we do not have a graph available.
In the previous example, we were provided with a graph, and finding the relative extrema was a visual task. However, we will not always be given the graph of a function. What can we do in these cases?
We can use what are known as the first and second derivative tests. These tests are based on Fermat's Theorem about stationary points.
Fermat's Theorem states that, if a function has a relative extremum at
and the function is differentiable at that point, then
.
The points where the derivative of a function is equal to 0 are called stationary points. The slope of the function at a stationary point is equal to 0.
If we look back at the example of the cubic function we can observe that the relative maximum and minimum are also points where the slope of the graph is equal to 0. Let's draw tangent lines at the relative extrema!
Fig. 5: Graph of a cubic function with tangent lines at its relative extrema.
There must be a link between derivatives and relative extrema.
Finding the stationary points is what is known as the First Derivative Test. A stationary point might be a local maximum or local minimum, or it might be neither. To determine this, we use what is known as The Second Derivative Test.
The second derivative test states that if is a function with a second derivative, and
is a stationary point, then:
In words, the Second Derivative Test tells us the following:
Let's try understanding this process with an example.
Find the local maxima and local minima of the function , if any.
Find the derivative of f using the Power Rule.
Evaluate at a critical point.
Apply Fermat's Theorem
Solve for c by factoring. Start by dividing the equation by 6.
Factor the left-hand side of the equation.
so
and
.
Find the second derivative of f.
Evaluate the second derivative at each critical point.
and
Since then there is a local maximum at
. Its value is
. Since
then there is a local minimum at
. Its value is
. Let's take a look at the graph of the function to see if this makes any sense.
Fig. 6: Graph of the cubic function showing its relative extrema.
We found the precise relative extrema of the function!
It is important to note that ifthe test becomes inconclusive. This might happen because graphs have points with a slope of zero that are not relative extrema. In such cases, it might be worth inspecting the graph of the function.
Find the relative extrema of the function .
Find the derivative ofusing the Power Rule.
Evaluate at a critical point.
Apply Fermat's Theorem.
Solve for c.
Find the second derivative of.
Evaluate the second derivative at the critical point.
Since we cannot conclude anything from these tests. Let's now take a look at the graph of the function:
Fig. 7: Graph of a cubic function with no relative extrema.
Note that this function does not have relative extrema, even when we found that its derivative at is equal to zero. This point is still critical because the slope of the function is equal to 0 at that point. Note that the function also does not have a global maximum or a global minimum!
Further information on the function can be obtained by finding more of its derivatives, assuming they exist. This is known as the higher-order derivative test.
Unfortunately, there is no formula for finding the maxima and minima of a function. Locating extrema depends completely on the type of function and the shape of its graph.
Looking at the graph of the function is always a good first step! For example, if the function is a parabola opening downwards you can find its global maximum by finding its vertex. If you need to find local maxima and local minima without a graph, you can use the first and second derivative tests that we explored above.
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What are maxima and minima?
Maxima and minima are the plurals of maximum and minimum. A maximum is the greatest output of a function. A minimum is the least output of a function.
How to find maxima and minima?
Relative maxima and minima of a function can be found by performing the first and second derivative tests.
What are relative maxima and relative minima?
A relative maximum of a function is an output that is greater than the outputs next to it. A relative minimum of a function is an output that is less than the outputs next to it.
What is an example of maxima and minima?
A parabola is a good example of a function with a maximum or a minimum. If it opens upwards, the vertex is a minimum. If it opens downwards, the vertex is a maximum.
What is the formula and rule for finding maxima and minima?
There is no formula for finding maxima or minima. It depends on which function is being studied. You can get some information by taking a look at the graph. You can also do the first and second derivative tests.
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