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

## Example Questions

Explanation:

Explanation:

Explanation:

Explanation:

### Example Question #301 : Derivatives

Tell whether f is concave up or concave down when . How do you know?

Concave up, because .

Concave down, because .

Concave up, because .

Concave down, because .

Concave up, because .

Explanation:

Tell whether f is concave up or concave down when . How do you know?

To test for concavity, we need to find the second derivative.

In this case, we can use the power rule to do all our differentiation.

Power rule:

We will use this on each term in order to find our first and then second derivative.

For each term, we will decrease the exponent by 1, and then multiply by the original exponent.

Do the same thing to f'(x) to get our final function.

Now, to test for concavity, we will plug in our x value (5), and look at the sign of the result.

SO, our  second derivative is positive, very positive.

This means that f(x) is concave up at x=5.

### Example Question #11 : Second Derivatives

Determine the intervals on which the function is concave down:

The function is never concave down

Explanation:

To determine the intervals on which the function is concave down, we must determine the intervals on which the function's second derivative is negative.

First, we must find the second derivative of the function:

The derivatives were found using the following rules:

Next, we must find the values at which the second derivative is equal to zero:

Using the critical value, we now create intervals over which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, while on the second interval, the second derivative is positive. Therefore, the function is concave down on the first interval,

### Example Question #11 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''

Determine the intervals on which the function is concave up:

The function is never concave up

The function is never concave up

Explanation:

To determine the intervals on which the function is concave up, we must determine the intervals on which the function's second derivative is positive.

First, we must find the second derivative of the function:

The derivatives were found using the following rules:

The second derivative is a constant and is negative, which means that the function is concave down on its domain, never concave up.

### Example Question #302 : Derivatives

A relative minimum of the graph of   can be located at (nearest hundredth):

The graph of  has no relative minimum.

The graph of  has no relative minimum.

Explanation:

At a relative minimum  of the graph , it will hold that  and

First, find . Using the sum rule,

Apply the Constant Multiple and Power Rules:

Set this equal to 0:

Both zeroes of  are imaginary. Since  has no real zeroes, it follows that the graph of  has no relative extrema - maximum or minimum - on the coordinate plane.

### Example Question #19 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''

Determine the intervals on which the function is concave down:

Explanation:

To determine the intervals on which the function is concave down, we must determine the intervals on which the function's second derivative is negative.

First, we must find the second derivative of the function,

which was found using the following rules:

Next, we must find the values at which the second derivative is equal to zero:

Note that we factored by grouping to solve for x.

Using these values, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, on the second interval, the second derivative is positive, on the third interval, the second derivative is negative, and on the fourth interval, the second derivative is positive. The function is concave down on the intervals where the second derivative is negative, .

### Example Question #11 : Relationship Between The Concavity Of ƒ And The Sign Of ƒ''

Determine the intervals on which the function is concave up:

Explanation:

To determine the intervals on which the function is concave up, we must determine the intervals on which the function's second derivative is positive.

First, we must find the second derivative of the function,

which was found using the following rules:

Next, we must find the values at which the second derivative is equal to zero:

Using these values, we now create intervals on which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is positive, on the second interval, the second derivative is negative, and on the third interval, the second derivative is positive. So, the function is concave up on the intervals