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. A function f is concave up (or upwards) where the derivative f, prime is increasing. This is equivalent to the derivative of f, prime, which is f, start superscript, prime, prime, end superscript, being positive. Similarly, f is concave down (or downwards) where the derivative f, prime is decreasing (or equivalently, f, start superscript, prime, prime, end superscript is negative).
Graphically, a graph that's concave up has a cup shape, \cup, and a graph that's concave down has a cap shape, \cap.
Want to learn more about concavity and differential calculus? Check out this video.

Practice set 1: Analyzing concavity graphically

Problem 1.1
Select all the intervals where f, prime, left parenthesis, x, right parenthesis, is greater than, 0 and f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, is greater than, 0.
Velg alle utsagnene som er korrekte:
Velg alle utsagnene som er korrekte:

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Practice set 2: Analyzing concavity algebraically

Problem 2.1
f, left parenthesis, x, right parenthesis, equals, 3, x, start superscript, 4, end superscript, minus, 16, x, start superscript, 3, end superscript, plus, 24, x, start superscript, 2, end superscript, plus, 48
On which intervals is the graph of f concave down?
Velg ett svar:
Velg ett svar:

Want to try more problems like this? Check out this exercise.