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Jan 30, 2013 · Sal introduces the concept of concavity, what it means for a graph to be "concave up" or "concave down," and how this relates to the second derivative of a function. Created by Sal Khan. Practice...
- 10 min
- 512.9K
- Khan Academy
Review your knowledge of concavity of functions and how we use differential calculus to analyze it. What is concavity? Concavity relates to the rate of change of a function's derivative.
- You have two functions f and g, where f''>0 and g''>0. If you take the second derivative of f+g, you get f''+g'', which is positive. So their sum i...
- In order for 𝑓(𝑥) to be concave up, in some interval, 𝑓 ''(𝑥) has to be greater than or equal to 0 (i.e. non-negative) for all 𝑥 in that inter...
- In general, no. Points where f"(x)= 0 are specifically called inflection points. But in the example provided, -2 is also a critical point (Observe...
- Product rule: d/dx[f(x)*g(x)] = f'(x)g(x) + f(x)g'(x) f(x) = (x-1)³ and g(x) = x
Step 1: Given f (x), find f (a), f (b), f (c), for x= a, b and c, where a < c < b. Where a and b are the points of interest. C is just any convenient point in between them. Step 2: Find the equation of the line that connects the points found for a and b. Step 3: Find that y-coordinate of the line from Step 2 at point C.
- 2 min
- Sal Khan
- Yes. However, it will often be less work (and less prone to error) to use the Calculus method. Anyway here is how to find concavity without calculu...
- Interesting question – I was sure this wasn't true, but it was harder than I expected to come up with a counter example. Try this: `f(x) = sin(eˣ)...
- Typically in mathematics and natural sciences, we don't deal too much with the 3rd derivative. I think the easiest way to understand the 3rd deriva...
- The second derivative is one of the higher order derivatives. One example is the second derivative of the position function, s(t), which is equival...
- First half is correct, but if the slope is negative and f"(x) > 0, then the slope still increases – this means it becomes *less* negative, i.e. bec...
- Yep! That's correct. The second derivative being negative only implies that the first derivative is decreasing (becoming less positive or more nega...
- No, f can be negative, decreasing, both, or neither on that interval. If f'' is positive, it means the graph is concave up (that is, if you fit a c...
Jul 26, 2016 · Start practicing—and saving your progress—now: https://www.khanacademy.org/math/ap-c... Sal finds the intervals where g (x)=-x_+6x_-2x-3 is concave down/up by finding where its second ...
- 9 min
- 78.1K
- Khan Academy
Dec 21, 2020 · The important \(x\)-values at which concavity might switch are \(x=-1\), \(x=0\) and \(x=1\), which split the number line into four intervals as shown in Figure \(\PageIndex{7}\). We determine the concavity on each.
One use in math is that if f"(x) = 0 and f"'(x)≠0, then you do have an inflection point. Unfortunately, there are cases where f"'(x)=0 that are inflection points so this isn't always useful, but if the third derivative is easy to determine (e.g. for a polynomial) then it is worth trying.
- 9 min
People also ask
How to find concavity without calculus?
How do you find a positive concavity?
What is concavity in differential calculus?
Why do we need a second derivative of concavity?
State the first derivative test for critical points. Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph. Explain the concavity test for a function over an open interval.
