This is a real analysis question. For each n ∈ N, suppose that the functions fn : [−2,2] → [0,1] are continuous and have the property that for all α ∈ [0, 1], fn(αx + (1 − α)y) ≤ αfn(x) + (1 − α)fn(y) for all x ∈ [−2,2] and y ∈ [−2,2]. Note: notation fn : [−2, 2] → [0, 1] means that the domain of fn is [−2, 2] and that the range of fn is contained in [0, 1]. The goal of this problem is to use the Arzela`-Ascoli Theorem to prove that there is a subsequence of {fn, n ∈ N} which converges uniformly on [−1, 1]. (a) Verify that the {fn, n ∈ N} is uniformly bounded. (b) Prove that {fn, n ∈ N} is equicontinuous on [−1, 1]. Hint. Start by showing that fn(x) − fn(y) / x−y is (i) a non-decreasing function of x and y, and is (ii) bounded for all x, y ∈ [−1, 1].
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
This is a
For each n ∈ N, suppose that the functions fn : [−2,2] → [0,1] are continuous and have the property that for all α ∈ [0, 1],
fn(αx + (1 − α)y) ≤ αfn(x) + (1 − α)fn(y) for all x ∈ [−2,2] and y ∈ [−2,2].
Note: notation fn : [−2, 2] → [0, 1] means that the domain of fn is [−2, 2] and that the range of fn is contained in [0, 1].
The goal of this problem is to use the Arzela`-Ascoli Theorem to prove that there is a subsequence of {fn, n ∈ N} which converges uniformly on [−1, 1].
(a) Verify that the {fn, n ∈ N} is uniformly bounded.
(b) Prove that {fn, n ∈ N} is equicontinuous on [−1, 1]. Hint. Start by showing
that
fn(x) − fn(y) / x−y
is (i) a non-decreasing function of x and y, and is (ii) bounded for all x, y ∈ [−1, 1].
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