Question

In calculus, to determine where a function is increasing or decreasing, one should:

A) Take the first derivative and find critical points
B) Take the second derivative and find critical points
C) Set the function equal to zero
D) Use the product rule

Answer 1

To determine where a function is increasing or decreasing, one should take the first derivative and find critical points, then analyze the sign of the derivative around these points.

Correct option is A

Step-by-step explanation:

In calculus, to determine where a function is increasing or decreasing, one should take the first derivative and find critical points. This is because the first derivative of a function gives us the slope of the tangent to the curve at any given point. If this derivative is positive over an interval, the function is increasing there; if it is negative, the function is decreasing.

More specifically, here's the process:

1. Take the first derivative of the function.
2. Find where this derivative equals zero or does not exist; these places are called critical points.
3. Analyze the sign of the first derivative on intervals around the critical points to determine where the function is increasing or decreasing.

This method corresponds with option A) from the question. Option B) involves the second derivative, which instead tells us about the concavity of the function, not directly whether it's increasing or decreasing. Option C) is not related to the question of increase or decrease, and option D) refers to the product rule, which is a technique for finding derivatives, not for analyzing function behavior in this context.