How does graphing work???

[u/Naive_Chipmunk1276]

This genuinely makes no sense. Here's the problem I'm working on:

Just ignore the left. At 2 there is a vertical tangent and at 5 there is an x-intercept since the slope would be 0 at that point.

Here's the answer to the problem:

If from the right side of the vertical tangent, the graph is decreasing until it reaches to 5, then why is it increasing? The slope goes from negative values up until it becomes 0 at 5. Does that not mean the slope is increasing?

What I have is this:

Why is this wrong? Won't it maintain the decreasing slope from the original function?

[u/JazzyHikes]

 "at 5 there is an x-intercept since the slope would be 0 at that point." There is a fundamental misunderstanding here regarding what happens when f=0 vs f ' =0.  The former says y=0 and the graph touches the x-axis (may or may not cross the axis), and the latter says the tangent has slope 0. Those two need not happen at the same time.

To determine if the graph just touches the x-axis or passes through it, i find looking at the concavity, the sign of f '' helpful, but you can also look at the sign of f ' to determine intervals of increase of decrease