Question

score on last try: 0.73 of 1 pts. see details for more. next question get a similar question you can retry this question belo the graph of y = f(x) is shown below. list the critical values of f(x): enter the list of these values se question help: video submit question

Explanation:

Step1: Recall critical value definition

Critical values of a function \( f(x) \) are \( x \)-values where \( f'(x) = 0 \) or \( f'(x) \) does not exist (e.g., corners, cusps, vertical tangents). For a graph, we look for local maxima, minima, or points where the slope is undefined (sharp turns).

Step2: Identify critical points from graph

- At \( x = -2 \): Local maximum (slope changes from positive to negative).
- At \( x = -1 \): Sharp turn (slope undefined, corner).
- At \( x = 2 \): Local maximum (slope changes from positive to negative).
- At \( x = 4 \): Local minimum (slope changes from negative to positive).
- At \( x = 6 \): Slope changes (from increasing to... or undefined? Wait, looking at the graph, \( x = 6 \): the function has a "corner" or change in slope? Wait, re-examining: the graph at \( x=-2 \) (peak), \( x=-1 \) (corner), \( x=2 \) (peak), \( x=4 \) (valley), \( x=6 \) (where the slope changes from increasing? Wait, no—wait, critical points are where derivative is 0 or DNE. Let's list the \( x \)-coordinates of peaks, valleys, and sharp turns:

From the graph:
- \( x = -2 \): local max (derivative 0)
- \( x = -1 \): sharp turn (derivative DNE)
- \( x = 2 \): local max (derivative 0)
- \( x = 4 \): local min (derivative 0)
- \( x = 6 \): where the function changes from concave? Wait, no—wait, the graph at \( x=6 \): the slope changes? Wait, looking at the graph, at \( x=6 \), the function has a point where the slope changes (from a flatter increase to steeper? Wait, no, critical points are where derivative is 0 or DNE. Let's check each:

- \( x=-2 \): peak (derivative 0)
- \( x=-1 \): corner (derivative DNE)
- \( x=2 \): peak (derivative 0)
- \( x=4 \): valley (derivative 0)
- \( x=6 \): where the function has a "corner" or change in slope? Wait, the graph at \( x=6 \): before \( x=6 \), the function is increasing with a certain slope, after \( x=6 \), still increasing but maybe different? Wait, no—wait, critical points are where the derivative is 0 or DNE. So the points are \( x=-2, x=-1, x=2, x=4, x=6 \)? Wait, no, let's re-express:

Wait, the graph:
- At \( x=-2 \): local maximum (derivative 0)
- At \( x=-1 \): sharp corner (derivative DNE)
- At \( x=2 \): local maximum (derivative 0)
- At \( x=4 \): local minimum (derivative 0)
- At \( x=6 \): where the function has a "kink" or change in slope? Wait, no—wait, the graph at \( x=6 \): the function goes from a curve to a steeper curve? Wait, no, critical points are where the derivative is 0 or DNE. So the \( x \)-values are \( -2, -1, 2, 4, 6 \)? Wait, let's check again:

Looking at the graph:
- \( x=-2 \): peak (critical)
- \( x=-1 \): corner (critical)
- \( x=2 \): peak (critical)
- \( x=4 \): valley (critical)
- \( x=6 \): where the function has a point where the slope changes (derivative 0 or DNE? Wait, at \( x=6 \), the function is increasing, but before \( x=6 \), it's increasing with a different slope? Wait, no—wait, the graph at \( x=6 \): the function has a "corner" or is it smooth? Wait, the graph at \( x=6 \): the left side of \( x=6 \) is a curve, right side is a curve, but at \( x=6 \), is there a sharp turn? Wait, no, maybe I misread. Wait, the graph: from \( x=4 \) to \( x=6 \), it's increasing, then from \( x=6 \) onwards, still increasing but maybe the slope changes? Wait, no—critical points are where derivative is 0 or DNE. So the correct critical values (from the graph's peaks, valleys, and sharp turns) are \( x = -2, x = -1, x = 2, x = 4, x = 6 \)? Wait, no, let's list each:

- \( x = -2 \): local max (deriv 0)
- \( x = -1 \): sharp turn (deriv DNE)
- \( x = 2 \): local max (deriv 0)
- \( x = 4 \): local min (deriv 0)
- \( x = 6 \): where the function has a "kink" (deriv DNE? Or deriv 0? Wait, no—wait, the graph at \( x=6 \): the function is increasing, but before \( x=6 \), it's increasing with a certain slope, after \( x=6 \), the slope is steeper? Wait, maybe \( x=6 \) is a critical point. Wait, maybe the correct critical values are \( -2, -1, 2, 4, 6 \).

Answer:

\( -2, -1, 2, 4, 6 \)