The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in
[link] , one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an interior point, in which case the absolute extremum is a local extremum. Therefore, by
[link] , the point
at which the local extremum occurs must be a critical point. We summarize this result in the following theorem.
Location of absolute extrema
Let
be a continuous function over a closed, bounded interval
The absolute maximum of
over
and the absolute minimum of
over
must occur at endpoints of
or at critical points of
in
With this idea in mind, let’s examine a procedure for locating absolute extrema.
Problem-solving strategy: locating absolute extrema over a closed interval
Consider a continuous function
defined over the closed interval
Evaluate
at the endpoints
and
Find all critical points of
that lie over the interval
and evaluate
at those critical points.
Compare all values found in (1) and (2). From
[link] , the absolute extrema must occur at endpoints or critical points. Therefore, the largest of these values is the absolute maximum of
The smallest of these values is the absolute minimum of
Now let’s look at how to use this strategy to find the absolute maximum and absolute minimum values for continuous functions.
Locating absolute extrema
For each of the following functions, find the absolute maximum and absolute minimum over the specified interval and state where those values occur.
over
over
Step 1. Evaluate
at the endpoints
and
Step 2. Since
is defined for all real numbers
Therefore, there are no critical points where the derivative is undefined. It remains to check where
Since
at
and
is in the interval
is a candidate for an absolute extremum of
over
We evaluate
and find
Step 3. We set up the following table to compare the values found in steps 1 and 2.
Conclusion
Absolute maximum
Absolute minimum
From the table, we find that the absolute maximum of
over the interval [1, 3] is
and it occurs at
The absolute minimum of
over the interval [1, 3] is
and it occurs at
as shown in the following graph.
This function has both an absolute maximum and an absolute minimum.
Step 1. Evaluate
at the endpoints
and
Step 2. The derivative of
is given by
for
The derivative is zero when
which implies
The derivative is undefined at
Therefore, the critical points of
are
The point
is an endpoint, so we already evaluated
in step 1. The point
is not in the interval of interest, so we need only evaluate
We find that
Step 3. We compare the values found in steps 1 and 2, in the following table.
Conclusion
Absolute maximum
Absolute minimum
We conclude that the absolute maximum of
over the interval [0, 2] is zero, and it occurs at
The absolute minimum is −2, and it occurs at
as shown in the following graph.
This function has an absolute maximum at an endpoint of the interval.
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
