Finding Maxima and Minima - Math
Card 1 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
Tap to reveal answer
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
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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
Tap to reveal answer
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.
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Consider the function
Find the maximum of the function on the interval .
Consider the function
Find the maximum of the function on the interval
Tap to reveal answer
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
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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
Tap to reveal answer
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.
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Consider the function
Find the maximum of the function on the interval .
Consider the function
Find the maximum of the function on the interval
Tap to reveal answer
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
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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
Tap to reveal answer
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.
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Consider the function
Find the maximum of the function on the interval .
Consider the function
Find the maximum of the function on the interval
Tap to reveal answer
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
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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
Tap to reveal answer
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.
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