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- Concavity describes whether a graph opens upward (concave up) or downward (concave down). It indicates whether the graph is curving upwards like an "U" shape or downwards like an "n" shape.
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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
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.
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, ∞).
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.
Apr 24, 2022 · If \( f''(x) \) is positive on an interval, the graph of \( y=f(x) \) is concave up on that interval. We can say that \(f\) is increasing (or decreasing) at an increasing rate. If \( f''(x) \) is negative on an interval, the graph of \( y=f(x) \) is concave down on that interval.
1.1 Definitions. Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have:
1. f: C!R is concave i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a: For strict concavity, the inequalities are strict. 2. f: C!R is convex i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a: For strictly convexity, the inequalities are strict. 2
