Is strictly convex function differentiable?
A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable. for all x and y in the interval. is strictly convex.
What is a strictly convex function?
Strictly convex function, a function having the line between any two points above its graph. Strictly convex polygon, a polygon enclosing a strictly convex set of points. Strictly convex set, a set whose interior contains the line between any two points.
How do you know if a function is strictly convex?
We can determine the concavity/convexity of a function by determining whether the Hessian is negative or positive semidefinite, as follows. if H(x) is positive definite for all x ∈ S then f is strictly convex.
Does strictly convex imply convex?
If the inequality holds strictly (i.e. < rather than ≤) for all t ∈ (0, 1) and x = y, then we say that f is strictly convex. is convex. These conditions are given in increasing order of strength; strong convexity implies strict convexity which implies convexity.
Is convex function continuous?
A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.
What are the properties of convex function?
It is easy to show the following properties of convex functions: • If the functions f, g: Rn → R are convex, then so is the function f + g. If f : Rn → R is convex and λ ≥ 0, then also the function λf is convex. Every linear (or affine) function is convex.
What is strictly convex preference?
So, in two dimensions, with strictly monotonic preferences, strict convexity says that if two consumption bundles are each on the same indifference curve as x, then any point on a line connecting these two points (except for the points themselves) will be on a higher indifference curve than x.
What is strictly concave function?
A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. Points where concavity changes (between concave and convex) are inflection points.
How do you check if a function is strictly concave?
Thus if you want to determine whether a function is strictly concave or strictly convex, you should first check the Hessian. If the Hessian is negative definite for all values of x then the function is strictly concave, and if the Hessian is positive definite for all values of x then the function is strictly convex.
Is Cobb Douglas concave?
If our f(x, y) = cxayb exhibits constant or decreasing return to scale (CRS or DRS), that is a + b ≤ 1, then clearly a ≤ 0, b ≤ 0, and we have thus the Cobb-Douglas function is concave if and M1 ≤ 0, M1 ≤ 0, M2 ≥ 0, thus f is concave.
Can a function be both convex and concave?
A linear function will be both convex and concave since it satisfies both inequalities (A. 1) and (A. 2). A function may be con- vex within a region and concave elsewhere.
How do you prove strict convexity?
(1) The function is strictly convex if the inequality is always strict, i.e. if x = y implies that θf ( x) + (1 − θ)f ( y) > f (θ x + (1 − θ) y). (2) A concave function is a function f such that −f is convex. Linear functions are convex, but not strictly convex.
Can a convex function be non-differentiable at countably many points?
A convex function is differentiable at all but countably many points. Let $f:Bbb RtoBbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, for example $f(x)=intlfloor xrfloor,dx$.
Are convex maps always left-differentiable and right- differentiable?
It’s fairly trivial to demonstrate that whenever f is a convex map (on an open set), then f is both left-differentiable and right-differentiable at every point of its domain. It’s also quite typically easy to show that both f l ′ and f r ′ are both increasing functions.
Why is the derivative of the sum of a convergent sum zero?
Since the sum is convergent (assuming that x ≤ y are points such that f is differentiable at x and y so that this makes sense), there can only be countably many values in the sum which are non-zero, and at all other points the oscillation is zero and so the derivative exists.