Applications of Derivatives

Front

What does it mean if f '(x) > 0?

Back

The function f(x) is increasing.

Front

What does a negative derivative indicate about a function?

Back

The function is decreasing.

Front

Find the derivative of f(x) = x^4 + 4x^3 - 2x^2.

Back

f'(x) = 4x^3 + 12x^2 - 4x.

Front

How do you determine the intervals where f(x) is decreasing?

Back

Find where f'(x) < 0.

Front

At what x-value does f(x) have a minimum?

Back

The minimum occurs at x = 1.

Front

What is the first derivative test?

Back

A method to determine local maxima and minima by analyzing the sign of f'(x).

Front

What is concavity in relation to derivatives?

Back

Concavity refers to the direction of the curve; f''(x) > 0 indicates concave up, f''(x) < 0 indicates concave down.

Front

What is an inflection point?

Back

A point on the graph where the concavity changes.

Front

How do you find critical points of a function?

Back

Set f'(x) = 0 and solve for x.

Front

What does it mean if f''(x) > 0?

Back

The function is concave up at that point.