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

## Example Questions

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

What is the concavity of at x=1?

Concave Down

Concave Diagonal

No Concavity

Concave Up

Concave Horizontal

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 #1 : Second Derivatives

Find the second derivative.       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 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''    Explanation: ### Example Question #4 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''    Explanation: ### Example Question #2 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''    Explanation: ### Example Question #1 : Second Derivatives    Explanation: ### Example Question #1 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''    Explanation: ### Example Question #8 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''    Explanation: ### Example Question #1 : Second Derivatives    Explanation: ### Example Question #10 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''      