Finding increasing and decreasing intervals on a graph
Given the function
in
[link] , identify the intervals on which the function appears to be increasing.
We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from
to
and from
on.
In
interval notation , we would say the function appears to be increasing on the interval (1,3) and the interval
Graph the function
Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.
Using technology, we find that the graph of the function looks like that in
[link] . It appears there is a low point, or local minimum, between
and
and a mirror-image high point, or local maximum, somewhere between
and
Graph the function
to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.
The local maximum appears to occur at
and the local minimum occurs at
The function is increasing on
and decreasing on
For the function
whose graph is shown in
[link] , find all local maxima and minima.
Observe the graph of
The graph attains a local maximum at
because it is the highest point in an open interval around
The local maximum is the
-coordinate at
which is
The graph attains a local minimum at
because it is the lowest point in an open interval around
The local minimum is the
y -coordinate at
which is
Analyzing the toolkit functions for increasing or decreasing intervals
We will now return to our toolkit functions and discuss their graphical behavior in
[link] ,
[link] , and
[link] .
Use a graph to locate the absolute maximum and absolute minimum
There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The
coordinates (output) at the highest and lowest points are called the
absolute maximum and
absolute minimum , respectively.
To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See
[link] .
Not every function has an absolute maximum or minimum value. The toolkit function
is one such function.
Absolute maxima and minima
The
absolute maximum of
at
is
where
for all
in the domain of
The
absolute minimum of
at
is
where
for all
in the domain of
Finding absolute maxima and minima from a graph
For the function
shown in
[link] , find all absolute maxima and minima.
Observe the graph of
The graph attains an absolute maximum in two locations,
and
because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the
y -coordinate at
and
which is
The graph attains an absolute minimum at
because it is the lowest point on the domain of the function’s graph. The absolute minimum is the
y -coordinate at
which is
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
