Define concavity.

The direction a curve opens; concave up faces upward, concave down faces downward.

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Define concavity.

The direction a curve opens; concave up faces upward, concave down faces downward.

What is a point of inflection?

A point where a function changes concavity.

Define concave up in terms of the first derivative.

The slopes of tangent lines are increasing, or $f'(x)$ is increasing.

Define concave down in terms of the first derivative.

The slopes of tangent lines are decreasing, or $f'(x)$ is decreasing.

How is concavity related to the second derivative?

Concave up: $f''(x) > 0$ . Concave down: $f''(x) < 0$ .

What is a possible point of inflection?

A point where $f''(x) = 0$ .

What must be true at a true point of inflection?

$f$ must change concavity and $f''(x) = 0$ .

What does a positive second derivative indicate?

The function is concave up.

What does a negative second derivative indicate?

The function is concave down.

How to find possible inflection points?

Set the second derivative equal to zero and solve for x: $f''(x)=0$ .

Explain how the second derivative relates to the rate of change of the first derivative.

The second derivative measures the rate at which the slope of the tangent line to the original function is changing.

Why is it important to check the concavity on both sides of a possible inflection point?

To confirm that the concavity actually changes at that point, making it a true inflection point.

Describe the relationship between $f'(x)$ , $f''(x)$ , and the shape of $f(x)$ .

$f'(x)$ indicates increasing/decreasing behavior, while $f''(x)$ indicates the concavity (curvature) of $f(x)$ .

Explain how to determine intervals of concavity.

Find where $f''(x) > 0$ (concave up) and $f''(x) < 0$ (concave down).

What is the significance of an inflection point?

It marks a change in the behavior of the curve, from bending upwards to bending downwards, or vice versa.

What does it mean for a function to be 'concave up'?

The function's graph is shaped like a cup, holding water.

What does it mean for a function to be 'concave down'?

The function's graph is shaped like a frown, spilling water.

If $f''(x) = 0$ , does this guarantee an inflection point?

No, it only indicates a possible inflection point. The concavity must change.

How does concavity relate to optimization problems?

Concavity helps determine whether a critical point is a local maximum or minimum.

Explain the difference between a local and global extremum.

A local extremum is a maximum or minimum within a specific interval, while a global extremum is the absolute maximum or minimum over the entire domain.

How to find intervals where $f(x)$ is concave up/down?

Find $f''(x)$ . 2. Find where $f''(x) = 0$ or is undefined. 3. Test intervals using test values.

How to determine if a point is an inflection point?

Find $f''(x)$ . 2. Check if $f''(x) = 0$ at that point. 3. Verify concavity change on either side.

Steps to determine concavity of $f(x)=x^4 - 6x^2$ ?

Find $f'(x) = 4x^3 - 12x$ . 2. Find $f''(x) = 12x^2 - 12$ . 3. Solve $12x^2 - 12 = 0$ , giving $x = \pm 1$ . 4. Test intervals $(-\infty, -1)$ , $(-1, 1)$ , $(1, \infty)$ .

How do you use the second derivative test to find local extrema?

Find critical points where $f'(x) = 0$ or is undefined. 2. Evaluate $f''(x)$ at each critical point. 3. If $f''(x) > 0$ , local minimum. If $f''(x) < 0$ , local maximum. If $f''(x) = 0$ , test is inconclusive.

How do you find the absolute maximum and minimum of a function on a closed interval?

Find critical points within the interval. 2. Evaluate the function at the critical points and endpoints of the interval. 3. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

How to find the first derivative of $f(x) = x^3 + 2x^2 - 5x + 1$ ?

Apply the power rule to each term: $f'(x) = 3x^2 + 4x - 5$ .

How to find the second derivative of $f(x) = 3x^2 + 4x - 5$ ?

Apply the power rule to each term: $f''(x) = 6x + 4$ .

How to find the possible inflection points of $f(x) = 6x + 4$ ?

Set $f''(x) = 0$ and solve for $x$ : $6x + 4 = 0$ , so $x = -\frac{2}{3}$ .

How to check if $x = -\frac{2}{3}$ is an inflection point for $f(x) = x^3 + 2x^2 - 5x + 1$ ?

Check the sign of $f''(x)$ on either side of $x = -\frac{2}{3}$ . If the sign changes, it is an inflection point.

How to determine the concavity of $f(x) = x^3 + 2x^2 - 5x + 1$ at $x = 0$ ?

Evaluate $f''(0) = 6(0) + 4 = 4$ . Since $f''(0) > 0$ , the function is concave up at $x = 0$ .

All Flashcards

Define concavity.

The direction a curve opens; concave up faces upward, concave down faces downward.

What is a point of inflection?

A point where a function changes concavity.

Define concave up in terms of the first derivative.

The slopes of tangent lines are increasing, or $f'(x)$ is increasing.

Define concave down in terms of the first derivative.

The slopes of tangent lines are decreasing, or $f'(x)$ is decreasing.

How is concavity related to the second derivative?

Concave up: $f''(x) > 0$ . Concave down: $f''(x) < 0$ .

What is a possible point of inflection?

A point where $f''(x) = 0$ .

What must be true at a true point of inflection?

$f$ must change concavity and $f''(x) = 0$ .

What does a positive second derivative indicate?

The function is concave up.

What does a negative second derivative indicate?

The function is concave down.

How to find possible inflection points?

Set the second derivative equal to zero and solve for x: $f''(x)=0$ .

Explain how the second derivative relates to the rate of change of the first derivative.

The second derivative measures the rate at which the slope of the tangent line to the original function is changing.

Why is it important to check the concavity on both sides of a possible inflection point?

To confirm that the concavity actually changes at that point, making it a true inflection point.

Describe the relationship between $f'(x)$ , $f''(x)$ , and the shape of $f(x)$ .

$f'(x)$ indicates increasing/decreasing behavior, while $f''(x)$ indicates the concavity (curvature) of $f(x)$ .

Explain how to determine intervals of concavity.

Find where $f''(x) > 0$ (concave up) and $f''(x) < 0$ (concave down).

What is the significance of an inflection point?

It marks a change in the behavior of the curve, from bending upwards to bending downwards, or vice versa.

What does it mean for a function to be 'concave up'?

The function's graph is shaped like a cup, holding water.

What does it mean for a function to be 'concave down'?

The function's graph is shaped like a frown, spilling water.

If $f''(x) = 0$ , does this guarantee an inflection point?

No, it only indicates a possible inflection point. The concavity must change.

How does concavity relate to optimization problems?

Concavity helps determine whether a critical point is a local maximum or minimum.

Explain the difference between a local and global extremum.

A local extremum is a maximum or minimum within a specific interval, while a global extremum is the absolute maximum or minimum over the entire domain.

How to find intervals where $f(x)$ is concave up/down?

Find $f''(x)$ . 2. Find where $f''(x) = 0$ or is undefined. 3. Test intervals using test values.

How to determine if a point is an inflection point?

Find $f''(x)$ . 2. Check if $f''(x) = 0$ at that point. 3. Verify concavity change on either side.

Steps to determine concavity of $f(x)=x^4 - 6x^2$ ?

Find $f'(x) = 4x^3 - 12x$ . 2. Find $f''(x) = 12x^2 - 12$ . 3. Solve $12x^2 - 12 = 0$ , giving $x = \pm 1$ . 4. Test intervals $(-\infty, -1)$ , $(-1, 1)$ , $(1, \infty)$ .

How do you use the second derivative test to find local extrema?

Find critical points where $f'(x) = 0$ or is undefined. 2. Evaluate $f''(x)$ at each critical point. 3. If $f''(x) > 0$ , local minimum. If $f''(x) < 0$ , local maximum. If $f''(x) = 0$ , test is inconclusive.

How do you find the absolute maximum and minimum of a function on a closed interval?

Find critical points within the interval. 2. Evaluate the function at the critical points and endpoints of the interval. 3. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

How to find the first derivative of $f(x) = x^3 + 2x^2 - 5x + 1$ ?

Apply the power rule to each term: $f'(x) = 3x^2 + 4x - 5$ .

How to find the second derivative of $f(x) = 3x^2 + 4x - 5$ ?

Apply the power rule to each term: $f''(x) = 6x + 4$ .

How to find the possible inflection points of $f(x) = 6x + 4$ ?

Set $f''(x) = 0$ and solve for $x$ : $6x + 4 = 0$ , so $x = -\frac{2}{3}$ .

How to check if $x = -\frac{2}{3}$ is an inflection point for $f(x) = x^3 + 2x^2 - 5x + 1$ ?

Check the sign of $f''(x)$ on either side of $x = -\frac{2}{3}$ . If the sign changes, it is an inflection point.

How to determine the concavity of $f(x) = x^3 + 2x^2 - 5x + 1$ at $x = 0$ ?

Evaluate $f''(0) = 6(0) + 4 = 4$ . Since $f''(0) > 0$ , the function is concave up at $x = 0$ .