Derivatives & Graph Behavior

Front

What does it mean if g''(x) < 0 at a point?

Back

The function g(x) is concave down at that point.

Front

If the first derivative f'(x) is positive, what can we say about the function f(x)?

Back

The function f(x) is increasing on that interval.

Front

What is a critical number of a function?

Back

A critical number is a value of x where the derivative f'(x) is either zero or undefined.

Front

What does f''(x) > 0 indicate about the behavior of f'(x)?

Back

The function f'(x) is increasing.

Front

What is a point of inflection?

Back

A point of inflection is where the function changes concavity, i.e., from concave up to concave down or vice versa.

Front

If g(x) is concave up, what can we say about g''(x)?

Back

g''(x) > 0 in that interval.

Front

What does it mean if a function is decreasing on an interval?

Back

The first derivative f'(x) is negative on that interval.

Front

How do you find the intervals of concavity for a function?

Back

By analyzing the sign of the second derivative f''(x).

Front

What is the relationship between the first derivative and the original function?

Back

The first derivative f'(x) gives the slope of the tangent line to the function f(x) at any point.

Front

What does it mean if f'(x) = 0 at a point?

Back

It indicates a potential local maximum, local minimum, or a point of inflection.