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In calculus, a concave function is defined as a function whose second derivative is negative or decreasing. In other words, the slope of the function is decreasing as you move from left to right. The graph of a concave function is a curve that is bowed downward, and it looks like a frown.
Sep 9, 2023 · Convex Functions. Examples: Logarithmic functions, negative exponential functions. Properties: Slope decreases as you move along the function. Holds the property f(tx + (1−t)y) ≤ tf(x) + (1−t)f(y) for 0 ≤t ≤1. Ways to Remember: A convex function looks like a valley.
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) + +.
Definition. 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.
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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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.
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 $c$.
