Finding Maxima and Minima - Math
Card 0 of 8
Consider the function
Find the maximum of the function on the interval 
.
Consider the function
Find the maximum of the function on the interval
Notice that on the interval 
, the term 
is always less than or equal to 
. So the function is largest at the points when 
. This occurs at 
and 
.
Plugging in either 1 or 0 into the original function 
yields the correct answer of 0.
Notice that on the interval
Plugging in either 1 or 0 into the original function
Compare your answer with the correct one above
The function 
is such that
When you take the second derivative of the function 
, you obtain
What can you conclude about the function at 
?
The function
When you take the second derivative of the function
What can you conclude about the function at
We have a point at which 
. We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
We have a point at which
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
Compare your answer with the correct one above
Consider the function
Find the maximum of the function on the interval .
Consider the function
Find the maximum of the function on the interval
Notice that on the interval , the term is always less than or equal to . So the function is largest at the points when . This occurs at and .
Plugging in either 1 or 0 into the original function yields the correct answer of 0.
Notice that on the interval
Plugging in either 1 or 0 into the original function
Compare your answer with the correct one above
The function is such that
When you take the second derivative of the function , you obtain
What can you conclude about the function at ?
The function
When you take the second derivative of the function
What can you conclude about the function at
We have a point at which . We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
We have a point at which
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
Compare your answer with the correct one above
Consider the function
Find the maximum of the function on the interval .
Consider the function
Find the maximum of the function on the interval
Notice that on the interval , the term is always less than or equal to . So the function is largest at the points when . This occurs at and .
Plugging in either 1 or 0 into the original function yields the correct answer of 0.
Notice that on the interval
Plugging in either 1 or 0 into the original function
Compare your answer with the correct one above
The function is such that
When you take the second derivative of the function , you obtain
What can you conclude about the function at ?
The function
When you take the second derivative of the function
What can you conclude about the function at
We have a point at which . We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
We have a point at which
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
Compare your answer with the correct one above
Consider the function
Find the maximum of the function on the interval .
Consider the function
Find the maximum of the function on the interval
Notice that on the interval , the term is always less than or equal to . So the function is largest at the points when . This occurs at and .
Plugging in either 1 or 0 into the original function yields the correct answer of 0.
Notice that on the interval
Plugging in either 1 or 0 into the original function
Compare your answer with the correct one above
The function is such that
When you take the second derivative of the function , you obtain
What can you conclude about the function at ?
The function
When you take the second derivative of the function
What can you conclude about the function at
We have a point at which . We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
We have a point at which
Plug in 0 into the second derivative to obtain
So the point is an inflection point.
Compare your answer with the correct one above