The graph of the first derivative 𝑓 prime of a continuous function 𝑓 is shown. State the 𝑥-coordinates of the inflection points of 𝑓.
Inflection points are points where there is a change in concavity, so a change from concave up to concave down or concave down to concave up. When a function is concave up, the pink lines show that the slope of the tangent line increases. And so the graph of its first derivative is increasing. When a function is concave down, again the pink lines show that the slope of the tangent line is decreasing. And so the graph of the first derivative is decreasing. And so at an inflection point, the first derivative changes from increasing to decreasing or decreasing to increasing.
So if we look at our graph we can see that the first derivative goes from increasing to decreasing at 𝑥 equals two, decreasing to increasing at 𝑥 equals three, increasing to decreasing at 𝑥 equals five, and decreasing to increasing at 𝑥 equals seven. So we can conclude that 𝑓 has inflection points at 𝑥 equals two, 𝑥 equals three, 𝑥 equals five, and 𝑥 equals seven.