5. (a) Show that if the functions f and g are log-convex, f+g is also log-convex....
8.1. Consider the problenm min f(x) (P) t. g(x)s0 where f and g are convex functions over R" and X CR" is a convex set. Suppose that x is an optimal solution of (P) that satisfies g(x")<0. Show that x is also an optimal solution of the problem min f(x) s.t. xX.
8.1. Consider the problenm min f(x) (P) t. g(x)s0 where f and g are convex functions over R" and X CR" is a convex set. Suppose that x...
3. Show that (a) the function g: R” → R, given by g(x) = ||2||2, is convex. (b) if f : RM → R is convex, then g:R" + R given by g(x) = f(Ax – b) is also convex. A here is an m x n matrix, and b ERM is a vector. You may use any of the results we covered in class (but the definition of convexity may be an easy way to do this, and gives...
Please answer the following question. Please show all your working/solutions. Optimality conditions in general frequently require us to relate slopes of different curves (for example, the slope of an indifference curve and the slope of a budget constraint). For this reason, we deal with a lot of derivatives. The logarithm has a particularly convenient form, so we will use it extensively in this course. Consider the function f(X) = a log(X) where a is a parameter with a > 0....
5 For the functions f(x) = (2,2 – 2x)e- and g(x) = 2.5 – 3.23 + 22:2 (1) dentify and classify any stationary points using the second derivative test. (1) Identify and classify any points of inflection using the sign diagram of the second deriva- tive, (i) Determine the intervals where the function is concave up and concave down.
1. (a) Give an example of functions f and g defined on R such that both f and g are discontinuous at every c, but the sum +9 is continuous at every (b) Give an example of functions and g defined on R such that g is not constant, / is discontinuous at everyx, but the composition gol is continuous at every
Definition 5.48. Let f,g:X + Y be functions and assume that Y is a set in which the following operations make sense. Then the following are also functions: 1. f + g defined by (f +g)(x) = f(x) + g(x) for all x E X 2. f - g defined by (f – g)(x) = f(x) – g(x) for all x € X 3. f.g defined by (fºg)(x) = f(x) · g(x) for all x E X f(x) = "147...
19-20-21
19. [5pts.Each] Find the derivative of the functions: sin x a. y = b. g(x)=e* cosh(x²) 20. [5pts.] ALSO Q3 f(x)= x - cos x, 0<x< 277. Find the intervals of increasing & decreasing and intervals of concave up & concave down. Do not graph but find all relative/local maximum & minimum and inflection points if any. 21. [5pts.] Find the dimension of the rectangle of the largest area that can be inscribed in a circle of radius r....
Suppose f and g are continuous functions such that g(7) = 5 and such that f is defined at x = 7. Assume also that lim 5 f(x) + f(x)g(x)] = 20. Find f(7). f(7) =
We have the following Limit Comparison Test for improper integrals: Theorem. Suppose f(x), g(x) are two positive, decreasing functions on all x > 1, and that lim f(x) =c70 x+oo g(x) Then, roo 5° f(x) dx < oo if and only if ſº g(x) dx < 00 J1 (a) Using appropriate convergence tests for series, prove the Limit Comparison Test for improper integrals. (Hint: Define two sequences an = f(n), bn = g(n). What can you say about the limit...
Given a true PRG G, show whether or not the following
functions
are necessarily PRGs. If true, prove it. If not give a
counterexample. Note that jj denotes
concatenation
G'(s) : = G (s) II G (s 1
IsI )
Problem 2 (30 points). Given a true PRG G, show whether or not the following functions are necessarily PRGs. If true, prove it. If not give a counterexample. Note that l| denotes concatenation.
Problem 2 (30 points). Given a true...