|х x.for some real numbers x and y. Find all ordered pars (r, y) such that...
(1 point) Consider the function f(x, y) = 4x+ + 8y. List all critical points of f(x, y). If there are none, enter "none". If there is more than one, enter a comma-separated list of ordered pairs, e.g., "(1,2), (3,4). (5,6)" Critical points are List all critical points of f(x,y) which are local maxima. If there are none, enter "none". If there is more than one, enter a comma-separated list of ordered pairs, e... "(1,2), (3,4), (5,6) Local maxima occur...
PROBLEM 7 Find all critical points of g(x, y) = 12 y - 6xy - 63/2x + 144x + y2 + 3x - 6. Input your answer as a sequence of one or more ordered pairs separated by commas. For example: (1,2), (3,4), (5,6)
Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) @ (x, y) = (u + x +1, v + y + 1) for all (u, v) and (x, y) in V. Define an operation of "scalar multipli- cation" by a® (x, y) = (ax, ay) for all a E R and (x,y) E V Under the two operations the set V is not a vector space....
Suppose R is the relation defined on all real numbers by for all real numbers x,y (xRy if |x-yl3) Then for real numbers x and y, xR2y iff
QUESTION 2 - 1 POINT Find the intersection points of the parabola y = -0? - 2 and the line-x+y= 4. Give your answer as two ordered pairs separated by a comma. For example, if you found that the solutions were (1,2) and (3, 4) you would enter (1,2), (3,4).
4 Let R2 be the set of all ordered pairs of real numbers equipped with the operations: addition defined by (21,02) (91, 92) = (21 41, 22 y2) and scalar multiplication defined by c(x1,22) = (cx1,Cx2), herece R is a scalar. Note that the operation addition here is non standard. Is R’ in this case a vector space ? (Justify your answer)
Given are n real numbers x(1), x(2), ..., x(n). Some of them are positive, some may be negative. The total sum is positive. Prove the following statement: There exists some index i such that all the following n sums are positive: х() x()xi+1) x() x+1)xi+2) х() + x(+1) + x(і+2) + ... х(і+n-1) Here "plus" and "minus" within the brackets are meant modulo n. Given are n real numbers x(1), x(2), ..., x(n). Some of them are positive, some may...
Prove that there is a unique ordered pair (x, y) where x and y are real numbers such that y=x’ and y=2x-1. (Be sure to prove both “existence" and "uniqueness.") (25 pts)
Show all work. 1. Find two nonnegative real numbers x and y such that x + y = 24 and x2 + y2 is maximized and show why it is a maximum using calculus. 3. The length of a rectangle is decreasing at a rate of 2 cm/sec, while the width wis increasing at the rate of 3 cm/sec. At the moment when the length / is 12 cm and the width wis 5 cm, find the rate at which...
Prove that x^2+xy+y^2≥0 for all real numbers, x and y. Find the values that result in equality.