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Support function. The support function of a set C C R is defined as We allow Sc(y) to take on the value too.) Suppose that C and D are closed convex sets in R". Show that C D if and only if their...
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C i...
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on C if and only if Vf(x*)-(x - x*)2 0 for all x in C.
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on...
Running average of a convex function. Suppose fR R is convex, with R+ S dom f. Show that its running average F, defined as F(a)-f(t) dt. dom F-R++ 2 0 is convex. You can assume f is differentiable....
Running average of a convex function. Suppose fR R is convex, with R+ S dom f. Show that its running average F, defined as F(a)-f(t) dt. dom F-R++ 2 0 is convex. You can assume f is differentiable.
Running average of a convex function. Suppose fR R is convex, with R+ S dom f. Show that its running average F, defined as F(a)-f(t) dt. dom F-R++ 2 0 is convex. You can assume f is differentiable.
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3r E R s.t. Vy R, A function f visits a function g when Vz E R,3y E R s.t. < y...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3r E R s.t. Vy R, A function f visits a function g when Vz E R,3y E R s.t. < y and lf(y)-g(y)| < We were unable to transcribe this imageBelow are three claims. Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, show...
2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) i...
please explain the steps you take
2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete: If (An)1 is a Cauchy sequence (with our semi- metric d) then there is a measurable set A EM such that lim, too d(An, A) 0.
2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete:...
Problem 3: In this problem, we show that the product of integrable functions is integrable. Take...
Problem 3: In this problem, we show that the product of integrable functions is integrable. Take any integrable function : [a, b] → R. Let C be a constant satisfying (2) SC for all re(a,b). (a) Show that 1/(x)? - SW' <2C\/(x) - S() for r, y € (a,b). (bi Prove that is integrable. (c) If and g are integrable functions (a, 6] R. show that fe is integrable as well. Hint: first consider the function ( + 9)?.)
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
Warm-Up: Subgradients & More (15 pts) 1. Recall that a function f:R" + R is convex...
Warm-Up: Subgradients & More (15 pts) 1. Recall that a function f:R" + R is convex if for all 2, Y ER" and le (0,1), \f (2) + (1 - 1)f(y) = f(2x + (1 - 1)y). Using this definition, show that (a) f(3) = wfi (2) is a convex function for x ER" whenever fi: R → R is a convex function and w > 0 (b) f(x) = f1(x) + f2(2) is a convex function for x ER"...
3. Consider the production function y= . Assume r 1. (a) Show that the production function...
3. Consider the production function y= . Assume r 1. (a) Show that the production function y(x) is concave. (b) Show that the inverse production function r(y) is convex (c) The price of y is p 10. Find the firm's total product TP(x), marginal product MP(a) and average product AP(a) (d) Find the firm's value of marginal product VMP(a), and value of average product VAP(r) (e) Assume w < marVAP (x) Find the firm's input demand curve r*(w) (f) Suppose...
all a,b,c,d 1. Suppose C is simple closed curve in the plane given by the parametric equation and recall that the outward unit normal vector n to C is given by y(t r'(t) If g is a scalar field on...
all a,b,c,d
1. Suppose C is simple closed curve in the plane given by the parametric equation and recall that the outward unit normal vector n to C is given by y(t r'(t) If g is a scalar field on C with gradient Vg, we define the normal derivative Dng by and we define the Laplacian, V2g, of g by For this problem, assume D and C satisfy the hypotheses of Green's Theorem and the appropriate partial derivatives of f...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of continuous functions from [0, 1 to R equipped with the usual function addition and scalar multiplication, is a vector space. (b) Let (V, K, +,-) be a vector space. Show that a non-empty subset W C V which is closed under and - necessarily contains the zero vector. (c) Is the set {(x,y)T: z,y E R, y a subspace of R2? Justify.
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on C if and only if Vf(x*)-(x - x*)2 0 for all x in C.
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on...
Running average of a convex function. Suppose fR R is convex, with R+ S dom f. Show that its running average F, defined as F(a)-f(t) dt. dom F-R++ 2 0 is convex. You can assume f is differentiable.
Running average of a convex function. Suppose fR R is convex, with R+ S dom f. Show that its running average F, defined as F(a)-f(t) dt. dom F-R++ 2 0 is convex. You can assume f is differentiable.
please explain the steps you take
2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete: If (An)1 is a Cauchy sequence (with our semi- metric d) then there is a measurable set A EM such that lim, too d(An, A) 0.
2. Let M be the set of all measurable sets in R, and let d be our semi-metric, show that (M, d) is complete:...
Problem 3: In this problem, we show that the product of integrable functions is integrable. Take any integrable function : [a, b] → R. Let C be a constant satisfying (2) SC for all re(a,b). (a) Show that 1/(x)? - SW' <2C\/(x) - S() for r, y € (a,b). (bi Prove that is integrable. (c) If and g are integrable functions (a, 6] R. show that fe is integrable as well. Hint: first consider the function ( + 9)?.)
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
Warm-Up: Subgradients & More (15 pts) 1. Recall that a function f:R" + R is convex if for all 2, Y ER" and le (0,1), \f (2) + (1 - 1)f(y) = f(2x + (1 - 1)y). Using this definition, show that (a) f(3) = wfi (2) is a convex function for x ER" whenever fi: R → R is a convex function and w > 0 (b) f(x) = f1(x) + f2(2) is a convex function for x ER"...
3. Consider the production function y= . Assume r 1. (a) Show that the production function y(x) is concave. (b) Show that the inverse production function r(y) is convex (c) The price of y is p 10. Find the firm's total product TP(x), marginal product MP(a) and average product AP(a) (d) Find the firm's value of marginal product VMP(a), and value of average product VAP(r) (e) Assume w < marVAP (x) Find the firm's input demand curve r*(w) (f) Suppose...
all a,b,c,d
1. Suppose C is simple closed curve in the plane given by the parametric equation and recall that the outward unit normal vector n to C is given by y(t r'(t) If g is a scalar field on C with gradient Vg, we define the normal derivative Dng by and we define the Laplacian, V2g, of g by For this problem, assume D and C satisfy the hypotheses of Green's Theorem and the appropriate partial derivatives of f...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of continuous functions from [0, 1 to R equipped with the usual function addition and scalar multiplication, is a vector space. (b) Let (V, K, +,-) be a vector space. Show that a non-empty subset W C V which is closed under and - necessarily contains the zero vector. (c) Is the set {(x,y)T: z,y E R, y a subspace of R2? Justify.
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