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1. Let x, a € R. Prove that if a <a, then -a < x <a.
(2.2) Let a be a real number with 1<a< 2. Put f(x) = Q +r 1+2 (a) Show that f maps (1, 0) into (1, 0). (b) Show that f is a contraction on [1, ) and find its fixed point.
Show that ze?-? = cosz has one solution on {2EC:[2]<1}, and the solution is real number.
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
Please Prove the Following: Prove that if A is a finite set (i.e. it contains a finite number of ele ments), then IAI < INI, and if B s an infinite set, then INI-IBI
5. Describe the following sets of real numbers and find the supremum and infimum of these sets: (a) {x}\x2 – 2 <4€R} (b) {x|x+ 2 +13 – x4<4} (©) {x|x<for all neN} 6. For any two elements x and y of an ordered field, prove that _x+ y + x- x + y - x - y (a) max{x,y}=- (b) min{x,y}=-
5. Find the absolute max and absolute min of f(:1, y) = x2 + 2y2 – 2.0 – 4y on the rectangle (<r<2,0 <y<3.
Find any global max or global min ) For the function f(x) = 2x3 - 6x2 +6 ;(-1<x<3)
4. Find & such that |--^x=12,< for all \x + 2<8.
2. A random variable X has a cdf given by F(x) = 1 . x < 0 0 < x < 1 <3 x > 3 11, (f) What is P(X = 1)? (g) Find E(X), the expectation of X. (h) Find the 75th percentile of the distribution. Namely, find the value of 70.75 SO that P(X < 70.75) = F(710.75) = 0.75. (i) Find the conditional probability P(X > X|X > 3).