Question

Use the graph of the function f

to fill in the blanks below.
(You can click on the graph to enlarge the image.)
The critical points of f(x)
on the open interval (−4,1) occur at x=
functions
equation

Answer 1

The critical points of \( f(x) \) on the open interval \( (-4,1) \) occur at \( x = -2 \) and \( x = 0 \).

Step-by-step explanation

In the given graph of the function \( f \), critical points are identified where the derivative is equal to zero or undefined. The open interval is specified as \( (-4,1) \), indicating that we are focusing on the values of \( x \) between -4 and 1, excluding the endpoints.

To find the critical points, we look for values of \( x \) where the derivative of \( f(x) \) is zero or undefined. This is because at these points, the slope of the graph is either flat (horizontal tangent) or undefined (vertical tangent). By analyzing the graph, it is evident that \( x = -2 \) and \( x = 0 \) are the points where the derivative is zero, making them critical points.

In summary, critical points are essential in understanding the behavior of a function. In this case, \( x = -2 \) and \( x = 0 \) are the critical points of \( f(x) \) on the open interval \( (-4,1) \), providing insights into the turning points or extremum of the function within this specific range.