This is the graph of the derivative of
𝑓
.

What are the x-values of the inflection points for
𝑓
?

Question

This is the graph of the derivative of 
𝑓
.

What are the x-values of the inflection points for 
𝑓
?

BlackTom AI · Accepted Answer

To locate inflection points of f, we look at where f'' changes sign. Since we are given the graph of f'(x), recall that f''(x) is the derivative of f'(x). Therefore, f'' changes sign where f'(x) has local extrema (peaks or valleys), i.e., where f'(x) switches from increasing to decreasing or from decreasing to increasing.

First, examine the graph of f'(x) and identify its turning points. There is a clear local minimum of f'(x) near x = -2. This is where f'(x) stops decreasing and begins to increase, indicating a change in the sign of f'' around that x-value.

Next, look for another turning point where f'(x) attains a local maximum. Around x = 1, f'(x) reaches a peak and then starts to decrease again, which again signals a sign change in f''.

These two x-values, where f'(x) has local extrema, are where the second derivative f'' changes sign, and hence where f has inflection points.

To confirm the proposed x-values, check that there are no other turning points on the graph of f' that would imply additional inflection points for f. The graph shows only two clear extrema, at roughly x = -2 and x = 1, with f' transitioning its slope at those points, not elsewhere.

Therefore, the inflection points of f occur at x ≈ -2 and x ≈ 1, corresponding to the x-values where f' has a local minimum and a local maximum respectively.

MATH-112-301-001 Unproctored Midcourse Exam 3 Practice Exam 2

This is the graph of the derivative of 𝑓 . What are the x-values of the inflection points for 𝑓 ?

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