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Dec 21, 2020 · In general, concavity can change only where either the second derivative is 0, where there is a vertical asymptote, or (rare in practice) where the second derivative is undefined. But concavity doesn't \emph{have} to change at these places.
Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
- 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
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.
Describe how the second derivative of a function relates to its concavity and how to apply the second derivative test. Describe the relationship between inflection points and concavity and how to find the inflection points of a function.
Dec 21, 2020 · If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
If f ′ (x) is positive on an interval, the graph of y = f(x) is increasing on that interval. If f ′ (x) is negative on an interval, the graph of y = f(x) is decreasing on that interval. The second derivative tells us if a function is concave up or concave down.
Given any x 1 or x 2 on an interval such that x 1 < x 2, if f (x 1) > f (x 2), then f (x) is decreasing over the interval. In the graph of f' (x) below, the graph is decreasing from (-∞, 1) and increasing from (1, ∞), so f (x) is concave down from (-∞, 1) and concave up from (1, ∞).
