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

Q: Find the number in the interval [0,3] such that the difference of its square from its cube is the…

A: Given problem:- Find the number in the interval [0, 2/3] such that the difference of its square from…

Q: Prove that if the equationf(x)=a0xn+a1xn−1+…+an−1x=0has a positive root x=x0, then the…

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Q: Calculate the real root of f(x)=x^3-2x^2-2x-3.4 with the first iteration of Regula Falsi (False…

A: 2.52 is the correct answer.

Q: Let f(x) = x2 + bx + c, b is negative odd integer, f(x) = 0 has two distinct prime number as roots,…

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Q: 1) Prove that there are no positive integer solutions to the equation x2-y?=1

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Q: Find two positive real numbers x and y such that they add to 120and x 2y is as large as possible.

A: Given: x+y=120and  y=120-xAlso given that z=x2ybut we have z=x2120-x=120x2-x3    ...(1)

Q: Find all positive integers n such that ϕ(n) = 12. Be sure to prove that you have found all…

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Q: (c) Prove that the equation x + 5x = 4cos.x cannot have more than one real root.

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Q: 2. Show that c(x, t) = solves 1 ac(x, t) 2 ax ac(x, 1) ar

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Q: Let a be a constant. Given that 3x In(ax)dr = el² ( 12 – 3 * (12- )- -23 find In a. 3 %3D

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Q: 1. Use the Intermediate Value Theorem to prove that the equation 3x - 4x? = 3 is solvable on the…

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Q: 1 Find the absolute maxima and minuma x²-2x4+ bunded by y= x², y=t, the closed negion on %3D…

A: Given that, f(x,y)=x2-2xy+y Take the derivative for x, fx=2x-2y Take the derivative for y, fy=-2x+1…

Q: "he pmf of X is 4.x – 1 - fx(x) = where x = 1, 2, 3 %3D %3D to

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Q: Show that f(x) = x4 + 4.x + c has at most two real roots for any umber c.

A: Consider the given equation. fx=x4+4x+c Case 1: When c>0. No sign change, by Descartes rule of…

Q: Find the infimum and supremum of the set X = {3 – Vztn ine N}. -

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Q: Find all positive integers x, y, z satisfying x · y · z² = 5xyz. '*

A: Introduction :Given , positive integers x,y,zwe have asked to find the values of x,y,z which…

Q: Prove using the IVT. 2x = bx has a solution if b > 2.

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Q: Solve the folowing equation on [0,2pi) 2sqrt2 +4cosx=0

A: In the given equation, the cosine function is the trigonometric function:   Rewrite the given…

Q: Prove that Σ = 2" . a. x=0 b. = 0. x=0

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Q: with Zeros of 2i with integer coeRficients that haf E Puno

A: We need to find the polynomial of degree three with integer coefficient

Q: Show that for any real constants a and b, where b > 0, (n + a)» = ©(nº)

A: To show for any real constants a and b where b > 0, n+ab=Θnb

Q: @[V2]is Principal Ideal Domain. O True O False

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Q: Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x2 +…

A: We already know, (x,y,z) = (3,4,5) is a solution of the equation, x² + y² = z² i.e. 3²+4² = 5² Now,…

Q: Q[V2]is Principal Ideal Domain. O True False

A: We have to solve given problem:

Q: If x,y,z are real numbers satisfying the equation 25 (9x* +y° )+9z²-15(5xy+yz+ 3zx )=0 then prove…

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Q: Prove that f(x) = 5x? + 3x – 26 is 0(x²). %3D

A: Given: f(x) = 5x2 + 3x – 26  is  θ(x 2).

Q: Is 1z = 1 for all z ∈ C?

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Q: there exists a real number s>0 such that |2x-5|<.01 wherenever |x-2.5|<s

A: We can prove that there exists S such that given condition holds. Actually we find a particular…

Q: Prove that the equation * + + = 1 has no solutions in positive integers x , Y, Z.

A: x/y + y/z + z/x = 1 if x , y , z are positive integers then case 1 :  If all three numbers are…

Q: Find the rank and nullity of the 2. 4 9. A z -7 2. -5 4 L7 -9 2 -4 verify the that dimension…

A: The given matrix is  A=20464-7201-5247-92-4 According to dimension theorem for any matrix A of size…

Q: Prove that the real part of is constant for -n < 0 < n z+2-i

A: Let  z = a+ib

Q: Find the two positive integers x and y such that x + y = 60 an 2 xy is maximum

A: The equation is x+y=60 where x and y are two positive integers. The main objective is to find x and…

Q: Use Cramer's rule to find only the value of x2 , where a is a positive real number. 2x2 – x3 + x4 =…

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Q: Assume that (8) ∞∑n=1nanxn−1=∞∑n=0anxn∑n=1∞nanxn−1=∑n=0∞anxn for all x, and determine what this…

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Q: Let a be a positive real number and I = , Sdydx. 1 2vx

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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: 2 (x) 2. prove that s

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Q: Find three positive integers x, y, and z that satisfy the given conditions. The product is 8, and…

A: Here, We have 3 positive integers x,y,z. Conditions are, x * y * z = 8…

Q: (a)Show that,if x is small enough for its cube and higher powers to be 1 neglected, = 1-x +. by…

A: It is given that x is small enough for its cube and higher powers to be neglected. It is required to…

Q: 6. Use the IVT to prove that x2+ In r = has a solution. %3D

A: The Intermediate value theorem says: If the continuous function attains two values then it must…

Q: Let ß be a positive real number. Evaluate L Bx-1 VxP + 3dx . (50+3 V50+3 - 2ß+3 /20+3 ㅇ 름(5v5-2v2) ㅇ…

A: We use substitution method to solve this integration

Q: 3 By applying the Intermediate Value Theorem choose the interval over which x4 = 7x3 - 5 will have a…

A: The given problem is to find the interval for the given function that which the intermediate value…

Q: 7. Use the Intermediate Value Theorem (IVT) to show that there is a root of the equation In r = 3 –…

A: We can use intermidiate theorem in order to solve the given problem.

Q: Q1. Prove that KT = = -.

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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: Prove that V/3 (1-x) (1+2x) 33

A: To Determine :-  ∫01 x13 1- x 231 + 2x  dx  = 2π33

Q: se Indirect Proof to prove that if a and b are both positive, and a2 < bi

A: