![]() ![]() Check if the second derivative changes signs before and after the inflection pointĢ. To prove whether or not the point is actually an inflection point, you can do two things:ġ. The second derivative test is used to find potential points of change in concavity (inflection points). ![]() Like the first derivative, the second derivative proves the first derivative's increase/decrease (if the second derivative is positive, the first derivative is increasing and vice versa). The second derivative itself doesn't prove concavity. The first derivative test is indeed used to prove the existence of critical points. The first derivative proves the function's increase/decrease (if the first derivative is positive, the function is increasing and vice versa). You seem to have gotten the second derivative stuff mixed up, so I'll just correct them (You've also missed some key terms in the first derivative) 7 5 > 0 h, prime, left parenthesis, 2, right parenthesis, equals, 3, point, 75, is greater than, 0 H h h h is increasing ↗ \nearrow ↗ \nearrow 5 ) = 1 5 > 0 h, prime, left parenthesis, minus, 0, point, 5, right parenthesis, equals, 15, is greater than, 0 ( − ∞, − 1 ) (-\infty,-1) ( − ∞, − 1 ) left parenthesis, minus, infinity, comma, minus, 1, right parenthesis H ′ ( x ) h'(x) h ′ ( x ) h, prime, left parenthesis, x, right parenthesis 7 5 > 0 f, prime, left parenthesis, 3, right parenthesis, equals, 0, point, 75, is greater than, 0 ( 0, 1 ) (0,1) ( 0, 1 ) left parenthesis, 0, comma, 1, right parenthesis 7 5 > 0 f, prime, left parenthesis, minus, 1, right parenthesis, equals, 0, point, 75, is greater than, 0į f f f is increasing ↗ \nearrow ↗ \nearrow Figure 3.F ′ ( x ) f'(x) f ′ ( x ) f, prime, left parenthesis, x, right parenthesis People who use applied mathematics, such as engineers and economists, often encounter the same types of functions where only small changes to certain constants occur. In the next part of our studies, we use calculus to make general observations about families of functions that depend on one or more parameters. Think about the Pythagorean Theorem: it doesn't tell us something about a single right triangle, but rather a fact about every right triangle. Mathematicians are often interested in making general observations, say by describing patterns that hold in a large number of cases. How can we construct first and second derivative sign charts of functions that depend on one or more parameters while allowing those parameters to remain arbitrary constants? Given a family of functions that depends on one or more parameters, how does the shape of the graph of a typical function in the family depend on the value of the parameters? Section 3.4 Using Derivatives to Describe Families of Functions Motivating Questions
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