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What does an increasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave up.

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What does an increasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave up.

What does a decreasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave down.

If the graph of $f''(x)$ is above the x-axis, what does this say about the concavity of $f(x)$ ?

$f(x)$ is concave up.

If the graph of $f''(x)$ is below the x-axis, what does this say about the concavity of $f(x)$ ?

$f(x)$ is concave down.

How can you identify a possible inflection point on the graph of $f''(x)$ ?

Look for points where the graph crosses the x-axis (i.e., where $f''(x) = 0$ ).

How can you identify intervals of concave up on the graph of $f'(x)$ ?

Look for intervals where the slope of $f'(x)$ is positive.

How can you identify intervals of concave down on the graph of $f'(x)$ ?

Look for intervals where the slope of $f'(x)$ is negative.

What does a horizontal tangent on the graph of $f'(x)$ indicate?

A possible inflection point on the graph of $f(x)$ .

If $f'(x)$ is a straight line with a positive slope, what does this indicate about $f(x)$ ?

$f(x)$ is concave up and has a constant rate of change of its slope.

If $f'(x)$ is a straight line with a negative slope, what does this indicate about $f(x)$ ?

$f(x)$ is concave down and has a constant rate of change of its slope.

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.

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: $f''(x)=0$ .