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The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below. Concave up.
Determine where the given function is increasing and decreasing. Find where its graph is concave up and concave down, the relative extrema, inflection points and sketch the graph of the function, A series of free Calculus Videos.
Theorem 3. Let C R be an open interval. 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:
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Sensitivity vs. specifity, cutoff selection, and ROC curves. How rare a disease is, and how it affects PPV and NPV. Interactive math video lesson on Concavity: Math-talk for "curvy" - and more on precalculus.
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.
- From that point on, the slope goes from being negative to becoming zero. Hence, it stops decreasing (in other words, it increases till it becomes z...
- Increasing slope can mean one of two things: _more_ positive or _less_ negative. Whichever situation you have, increasing slope always implies conc...
- Short answer: no. Since the function f is not defined by some formula, only by the graph sal draw, you cant say wether or not these are parabolas....
- A point where both f''(x) = 0 and f''(x) changes sign (i.e. f(x) changes concavity) is called a point of inflection of f(x). Visually, the graph of...
- Observe that he was looking at the graph of f'(x). This graph is the derivative of the first graph. So, in the first graph, at that point, the deri...
- When f'(x) is positive, f(x) increases When f'(x) is negative, f(x) decreases When f'(x) is zero, it indicates a *possible* local max or min (use t...
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: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
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The mathematical definition of a function being concave between points $x_1$ and $x_2$ is the following: $\lambda f(x_1)+(1-\lambda)f(x_2) \leq f(\lambda x_1+(1-\lambda)x_2)$, for any $0 \leq \lambda \leq 1$. Can someone give a detailed, intuitive explanation of this theorem?
