5. (10 points) Prove that for all positive real numbers r and y, one has the inequality > 2 (u+ 112 < 4.

Q: 8. The domain of - consists of numbers x for which g(x)_0 that are in the domains of both – and,

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Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

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Q: Show that there is a solution to the following equation in the interval (2,3). (Hint: Use the…

A: Since there are two questions posted, we will answer the first question (1). If you want the second…

Q: Determine the positive real numbers x and y such that • x + y = 4 and 2.n2 + y² is small as…

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Q: The sum of two positive real numbers x and y is 6. Find an x and y among these such that the product…

A: We need to find x and y.

Q: Show that x3 +x – 1 has a root in the interval [0,1]. (R

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Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: f(x) = x4 + x - 6 ƒ is continuous over the closed interval [a, b] if and only if it's continuous on…

Q: 9. For |x|<1, prove that 8x 16x 24x 1-x1+x 1-x 2 8x 8x = 1+ (1 – x)? (1+x2 (1-x)? ...

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Q: Verify that f(x)=x3- 3x + 2 satisfies the hypotheses of the Mean Value Theorem on the interval…

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Q: Determine two unique values of θ in the interval [0,2π), such that                    sec (θ) = 2√3/…

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Q: 1) Verify that the function f(r) = r - 3x + 2 satisfies the hypotheses of the Mean Value Theorem on…

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Q: 7. Verify that the function f(x) =x3-x'-6x+2 satisfies the hypotheses of Rolle's Theorem on the…

A: Given:Verify that the following function satisfies the hypotheses of Rolle's Theorem on the interval…

Q: Prove the inequality tanx2tanx1&gt;x2x1 provided 0&lt;x1&lt;x2&lt;π2.

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Q: 1 1. -to prove that x+y|# | x|+ | y| for all real numbers d) Use x = and y = %3D x and y.

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Q: Show that there are an infinite number of rational numbers between any two real numbers x and y with…

A: using Archimedean property to prove above therom...

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

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Q: find the interval of existence in which the particular solutions of the IVP is vaild

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Q: 1. prove that if x is real number then ,[2x] =[x] +[x +1/2] when [x] denote the greatest integer…

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Q: An arbitrary point x of 0.1 is chosen. Since it is known that x> = 5, what is the existence of x> =…

A: x is an arbitrary point of (0,1)  and x ≥0.5x≥78=0.8

Q: The sum of two positive real numbers x and y is 12. Find an x and y among these such that the…

A: The sum of two positive real numbers x and y is 12.

Q: . Show that there exists a real number 0 <x< 2 so that x' + 8x² – 10 = 0 -

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Q: Prove that there exists a unique real number solution to the equationx +x² – 1 = 0 between x = 2/3…

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Q: 6. Show that the half open interval [0, 2) is uncountable. (Hint: We know that (0, 2) is uncount-…

A: Since the interval (0, 2) contains the infinite subset {1.2,1.3,1.4,...} So (0, 2) is either…

Q: express the set of numbers x satisfying the given condition as an interval.       I 3x + 5 I &lt; 1

A: To solve the inequality: 3x+5&lt;1 Solution: We know that if x&lt;a then -a&lt;x&lt;a. We have…

Q: 4) Find the intervals where the function g is positive and enter exact answers only in interval…

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Q: Suppose x is a real number. Prove that if x 2, then there is a real number y such that x = (2y + 1)/…

A: Domain of a Function The domain of a function explains what values the independent variable, X, can…

Q: Verify that the function f(x) = x3 – 2x² – 4x + 2 satisfies the hypotheses of Rolle's Theorem on the…

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Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: Since x^4+x-8 is a polynomial it is continuous.Answer: f(x)=x^4+x-8 is continuous on the closed…

Q: Prove the inequality √1+x≤1+x2 for x&gt;1.

A: Consider the function. Find its derivative.

Q: 2) Use Rolle's Theorem on f(x) = 2x3 - 6x² - 2x + 6 to find all numbers c in the interval (1, 3) for…

A: Here is the solution:

Q: Find a positive number x such that the sum of 9x and is as small as possible. x = Does this problem…

A: Let, the sum be denoted by f(x) So, f(x) = 9x + 1/x  So, f'(x) = 9 - 1/x² &amp;, f"(x) = 2/x³

Q: Show that the IVP x" = –6x(x')3, x(-1) = -1, x'(-1) = 1/3, has a unique solution x and find the…

A: The given initial value problem is, x''=-6x(x')3                                     ..............1…

Q: What is the domain of { (z,3z2- 5) Iz ER except 0}? All real numbers All negative numbers {x\x€R>0}…

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Q: The domain of g(x) =3/(x - 2)consists of all real numbers except 2,represented in interval notation…

A: To Determine: The domain of g(x) interval notation as (- ∞, 2) ∪________. Given: we have a function…

Q: (1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute…

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Q: 1 Express the domain of f(x) using interval notation. Use oo for oo and use U to combine 2 - 2 – 3x…

A: We have to find the domain of the given function.

Q: 5. The set of all pairs of real numbers of the form (x, y), where x ≥ 0, with the standard…

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Q: Show that the interval [0, 1] is uncountable.

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Q: True or False The distance between two distinct points on the real number line is always greater…

A: Given: The distance between two distinct points on the real number line is always greater than zero.

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

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Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

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Q: Find two positive real numbers x and y such that they add to 123 and  x2y  is as large as possible.…

A: We have to find out x and y

Q: Find the maximum value of f(x)=x^3-3x on the set of all real numbers x satisfying x^4+36< (less then…

A: Given: The function fx=x3-3x To find: The maximum value of given function on the set of all real…

Q: Verify that the function f(x) = x² – 8x + 5 satisfies the three hypotheses of Rolle's Theorem on the…

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Q: 8) Find what va lues of x salisfy the fellowing inequality: 3x -1 >2 2x +5

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Q: Find all integer x satisfying the condition x gcd(4x +3, 2x). Justify your answer. %3D

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Q: The Intermediate Value Theorem can be used to prove that the equation 2²-2-15-0 has at least one…

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Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

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Q: 1 1 2 x y x+y || +

A: