AP Calculus AB : Relationship between the concavity of ƒ and the sign of ƒ''

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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Example Question #1 : Second Derivatives

What is the concavity of  at x=1?

Possible Answers:

Concave Diagonal

No Concavity 

Concave Down

Concave Up

Concave Horizontal

Correct answer:

Concave Up

Explanation:

First, find the double derivative of the function given.

You should get .

Next, plug in x=1 to get .

The double derivative is a positive number, which means the concavity is Concave Up. 

Example Question #2 : Second Derivatives

Find the second derivative. 

Possible Answers:

 

 

Correct answer:

 

Explanation:

We need to find the second derivative. To do this we will use power rule twice. We will take the first derivative by power rule, then the second derivative by power rule once again. Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent. 

Therefore, the first derivative is  and the second derivative is  by power rule.

Example Question #3 : Second Derivatives

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Example Question #4 : Second Derivatives

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Example Question #5 : Second Derivatives

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Example Question #1 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''

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Example Question #2 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''

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Example Question #3 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''

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Example Question #9 : Second Derivatives

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Example Question #10 : Second Derivatives

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