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.

## AP®︎/College Calculus BC

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

Lesson 9: Sketching curves of functions and their derivatives

# Curve sketching with calculus: polynomial

Sal sketches a graph of f(x)=3x⁴-4x³+2 including extremum and inflection points. Created by Sal Khan.

## Want to join the conversation?

• How can the slope be zero if the graph is concave upwards? My brain hurts :(
• Concave up means that the slope is increasing. If the slope was below zero and now it's zero, it has increased because 0 is a higher number than a negative number.
• what is the difference between a transition point and an inflection point?? I seem to have missed out on some of the things that Sal mentioned in the earlier lectures... Please help!
• There is no difference in this case. An inflection point (or point of inflection) is the point at which the concavity of the graph changes sign. In this case, the second derivative test is inconclusive, meaning that we must use a difference scheme to determine if x = 0 is in fact an inflection point.
• Why did Sal need to take the second derivative to find the inflection points? Couldnt he have used the first derivative? Im a bit confused.
• Inflection points are where the first derivative has relative max/mins (where the slope of the tangent line of the first derivative =0). He could have used the first derivative but not easily if he did it analytically. You can find points of inflection by looking at the graph of the first derivative, or by solving the 2nd derivative. (At least as far as I know...)
• how does a function look like (in graph) when the slope of a function is 0 and acceleration is 0?
I just cannot visualize it. ;o;
• A function that 0 slope and 0 concavity (acceleration) is just a horizontal line.
• Instead of the lengthy discussion starting at , could the third derivative have been used at 2/3 to determine that the second derivative changes sign in the way discussed?
• I know this is a year late but, for future viewers, yes you could have. If the third derivative at 2/3 is not zero, you would know that it's an inflection point. Furthermore, if f'''(2/3) was positive (which it was) you would know that the slope is concave downwards below that point and concave upwards above that point. (You would know that the slope of the second derivative at that point is positive)
• How can I evaluate the derivative of a function at an indicated point?
• You just take the derivative of that function and plug the x coordinate of the given point into the derivative.

So say we have f(x) = x^2 and we want to evaluate the derivative at point (2, 4). We take the derivative of f(x) to obtain f'(x) = 2x.

Afterwards, we just plug the x coordinate of (2,4) into f'(x). So, what we get is basically f'(x) = 2*2 which equals 4, and that's the slope of the original f(x) at the point (2, 4). The y coordinate, if they give you one, is just extraneous information, it's not needed in this problem.
• Guys I need some help with this: if you graph ((x+1)*lnx)/(x-1), the graph shows that when x=1, y=2, but why is it not undefined? This is so confusing for me.
• That's because the discontinuity is a "point discontinuity" or a "removable discontinuity". Graphical programs usually don't display this kind of discontinuities.
• how do you get the x and y intercept?
• To find the y-intercept, you make all x-values equal to 0 and solve for y. Inversely, to fine the x-intercept, you make all y-values equal to 0 and solve for x.
• If f''(c)=0, is c always an inflection point?Any example if otherwise?
• No, it is not always an inflection point.
If for example
f ''(x) = 3x^2
Then
f ''(0) = 0
But it's not an inflection point because you don't get a sign change at that point.
Notice that f ''(x) is positive on both sides of x = 0.
f ''(-1) = 3
f ''(1) = 3

So you can't assume that it will be an inflection point without checking for a sign change.