2nd Derivative Applications & Derivative Graphs

Front

What does it mean if f''(x) < 0 over an interval?

Back

f(x) is concave down over that interval.

Front

How do you determine points of inflection for a function g(x)?

Back

Points of inflection occur where the second derivative g''(x) changes sign.

Front

What indicates that a function f(x) is increasing?

Back

f'(x) > 0 (positive) indicates that f(x) is increasing.

Front

What is the relationship between a graph being increasing and its derivative?

Back

When a graph is increasing, its derivative f'(x) is positive.

Front

What is the relationship between a graph being decreasing and its derivative?

Back

When a graph is decreasing, its derivative f'(x) is negative.

Front

What does a positive second derivative f''(x) indicate about the concavity of f(x)?

Back

f(x) is concave up.

Front

How can you find the intervals where a function is concave up?

Back

Find where f''(x) > 0.

Front

What is the significance of critical points in the context of derivatives?

Back

Critical points occur where f'(x) = 0 or f'(x) is undefined, indicating potential local maxima or minima.

Front

What does it mean if a function has no points of inflection?

Back

The concavity of the function does not change over its domain.

Front

How do you determine if a function is decreasing on an interval?

Back

If f'(x) < 0 (negative) on that interval, then f(x) is decreasing.