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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
Concavity refers to the direction of the curvature of a function's graph. It indicates whether the graph is bending upwards (concave up) or downwards (concave down) and is closely related to the second derivative of the function.
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) + +.
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.
It is also possible to characterize concavity or convexity of functions in terms of the convexity of particular sets. Given the graph of a function, the hypograph of f,
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We make these ideas concrete with the following definitions: For a function $f$ that is differentiable at $c$, we say $f$ is concave up at $c$ means that $f(x)$ lies above the tangent line to $f$ at $x=c$, for all $x \ne c$ sufficiently near $c$.
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 $0 \leq \lambda \leq 1$.
