If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: AP®︎/College Calculus AB>Unit 5

Lesson 6: Determining concavity of intervals and finding points of inflection: graphical

# Inflection points (graphical)

Sal analyzes the graph of a function g to find all the inflection points of g.

## Want to join the conversation?

• I'm confused about the slope increasing and decreasing. Graphically, it looks like the slope changes signs around -3,0, and 3. Instead, the video showed the points of inflection in the middle of the slopes. Could you explain why that is in more depth?
• We are now caring about the slope of derivative. Or in other words, how the derivative of the derivative behaves.
Try changing "t" variable at the left here https://www.desmos.com/calculator/anqtkx2xtg
The red graph is your function (you can play with it)
The blue graph is your derivative
The green line is the slope of your derivative
• how does it become apparent that a slope is increasing or decreasing?
• An interesting trick that one can use for this is to draw the graph of the first derivative. Then identify all of the points in say f'(x) where the slope becomes zero. These points, where slope is zero are the inflection points. Instead of microanalyzing the graph for increasing or decreasing this is much more accurate and rigorous.
• i noticed something, as the slope increases the angle of the slope goes anti-clockwise and as the slope decreases angle of the slope goes clock-wise. And the point at which the 'clock' starts moving from anti-clockwise to clock-wise or vice versa, it is called point of inflection. Has anybody else noticed this?
• That's a really neat observation. it also links up with what positive and negative angles are, in relation to positive and negative slope.
• At , Sal states and writes there are 3 inflection points. I see 4 inflections points. Didn't he miss an inflection point at x = - 3.5? Over the interval (-4, -3.5) I estimated a slope of at least 18 and a slope of 10 over (-3.5, -3) ===> the slope is decreasing as the function enters a concave downward. Can anyone provide a counter argument?
• I don't know how to find the exact point. even sal says "around here"
• This is just finding inflection points graphically. To find the exact point you need the equation of the function and find the 2nd derivative. That will be explained in a later video. This seems to be just to get you comfortable with the concept of an inflection point.
• So between any 2 critical points there is always an inflection point?
• How exactly do you identify whether or not the slope is increasing or decreasing?The inflection point around the x value of -2 doesn't seem to have a change in slope from decreasing to increasing.
• The first derivative is the slope. when the derivative is 0 (a crititical point) the slope is 0.

The second derivative is the slope of the slope, or in other words if the slope is increasing or decreasing. when the second derivative is 0 (a critical point) the slope of the slope is 0, or in other words the slope goes from increasing to decreasing or the other way around. this does not mean the slope goes from positive to negative though (or the other way around.

You can have an increasing slope that is still negative, if the slope goes from -4 ot -3 it is still negative but increasing.

Let me know if that doesn't make sense.