Question

Let f(x) be a function whose derivative is shown below. a) On what intervals is f(x) increasing? On what intervals is f(x) decreasing?

Answer 1

The function f(x) is increasing where the derivative is positive and decreasing where the derivative is negative.

Step-by-step explanation:

The intervals where f(x) is increasing correspond to the values of ( x ) where the derivative f'(x) is positive. This is because a positive derivative indicates that the function is "going up" or increasing.

Conversely, the intervals where f(x) is decreasing align with the values of ( x ) where the derivative f'(x) is negative. A negative derivative signifies that the function is "going down" or decreasing.

Understanding the behavior of the derivative provides insights into the behavior of the original function. For instance, identifying critical points where the derivative is zero can pinpoint locations of maximum or minimum values for the function f(x) . Analyzing the sign changes in the derivative helps in delineating intervals where the function exhibits different behaviors.