(d) Using Part (c) and continuity of g, prove that g(x) = mx + l for some m, l e R. (e) Using Equation (1) and Part (d), prove that there exist a, b, c e R such that f(x) ax? + bx + c.

Q: 3) Let f(T) e R[T] be a polynomial of degree > 3. Show that there exists polynomials g(T) and h(T)…

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Q: 7. A function f : (a,6) → R satisfies a Lipschitz condition at x € (a,6) iff there is M > 0 and e >…

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Q: snip

A: The given statement is true. Because the local maximum is the highest point on the curve. so that…

Q: (a) Let f(x) = px² + 2qx +r with p > 0. By considering the minimum, prove that f(x) > 0 for all real…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: 2. Let f(x) = 2x²-3x+1 ,x 0.5 Determine constants A and B such that is continuous for all values of…

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Q: Consider the piecewise function 2+x² x<0 x=0 g(x) = 1, 2-³, 0<x< 1 2-x x 21 (a) For each of the…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Discuss the continuity of f(x, y, z) = 1 / (x2 + y2 − z) .

A: Here The function is discontinuous when the denominator is zero. Thus, That implies

Q: Q2) homeomorphism. Prove or disprove: The function f: (R, T1) - (R, 13) defined by f(x) = 3x is a…

A: Clearly the given function f(x)=3x is continuous. Now we want to check whether it is bijective,

Q: Let f(x)=x² -x² +x + 1 and g(x)= 3-х max{f(t):0sx; for 0 s x s 1} for 1 < x s 2 Discuss the…

A: Topic:- Functions

Q: lim r" use Cauchy Criteria to prove that exists and 0<r<1 n → 00

A: To Prove - Use Cauchy Criteria to prove that lim rn exists and 0 ≤ r &lt; 1 , n →∞         Theorem…

Q: a) Use Backward difference to find the Finite difference equation of the follow ди ди o'u -= C- xe…

A: We are given the equation ∂u∂t+a∂u∂x=c∂u2∂x2 , x∈ 0 , 1  , t≥0 Let us consider a grid of points in…

Q: Using continuity prove that f : (0, ∞) → R  defined by ƒ(x) : = 1/x is continous.

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Q: X'0, X1,* , Xn E (a, b) are arbitrary points such that x; # x; for i # j. Fur- thermore, let P be a…

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Q: Using the e, d definition of uniform continuity: Prove that f(x) = x³ is uniformly continuous on…

A: Given : fx=x3, -2&lt;x&lt;2 Let ε=δ12&gt;0 be given and for any arbitrary point a∈-2,2 we have…

Q: Questions 25-31.

A: Given, Theorem 1: Every polynomial function is continuous. Theorem 2: If two functions f(x) and…

Q: 2. Let f(x) = V4 – x for x < 4 and g(x) = x² for all x E R. (a) Give the domain of f · g, ƒ + g, ƒ º…

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Q: 5 - x, -1 sx 2 r - 1, 2 <xs 3 Discuss the continuity of g(x) = %3D

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Q: 3.x2 Let g(x) = 9. x > 0 -2x2 x <0 (a) Prove that g is differentiable on R.

A: A f(x) function f is differentiable at x iflim h→0 f(x+h)−f(x)h exists. i.e. if lim h→0+…

Q: (a) Let f: R" → R be a function given by f(x₁, 2,...,xn) = x²₁₁x²2x², where n = 1. Show that the…

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Q: just d

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Q: Using residue theorem to evaluate f (z+1)2 zp

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Q: 1. Let A= [x ∈ r: x ≠ 1/2] and B= [y ∈ r: ≠ -(1/2)] where r is the set of real numbers. given that…

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Q: Consider the function f : [−2, 6] → R, x→ f(x) = 2x3/ [√(x2+8)+2]. i) provide argument why this…

A: This is the question of Real topology, and real analysis.

Q: Let f(x) = x³ – 2x2 – 4x + 2 on I = [-2, 2]. Verify that f satisfies the conditions of Rolle's…

A: Rolle's Theorem1.f(x) is continuous in the interval [-2,2] because it is a polynomial function2.f(x)…

Q: b. Test the continuity of the following function 1, x< 0 V1-x, 0sxs1 ,atx 0,1 x- 2, x21

A: We have to check the continuity of f(x) at x=0, 1

Q: lim x – x has indeterminate form of type а. O b. 0 – ∞ | c. 0. 00 d. 00 + o е. 12 8 8|8

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Q: (a) Let g : [0, a] → R be differentiable, g(0) = 0, and|g'(x)| ≤ M for all x ∈ [0, a]. Show  |g(x)|…

A: Given as g : [0, a] → R be differentiable, g(0) = 0, and|g'(x)| ≤ M for all x ∈ [0, a] Use property…

Q: Let f : R −→ R be defined by f(x) = 2 − 5x. Show that f is bijective, and find (f^(−1))(x).

A: A function f is bijective if f is one- one  and onto

Q: Show that if n is a natural number and a, ß are real numbers with 8 > 0 then there exists a real…

A: Given that n is a natural number and α,β are real numbers with β&gt;0, then there exists a real…

Q: Show using Definition 4.3.1(Continuity) that if c is an isolated point ofA ⊆ R, then f : A → R is…

A: If c∈A⊆ R is an isolated point of A ⊆ R. To prove  f : A → R is continuous at c. Let ∈&gt;0 , Since…

Q: Let n be a natural number and let y1, Y2, Y3, . . . , Yn be real numbers. Prove by induction that…

A: (a) Consider n∈ℕ, and y1,y2,…,yn are real numbers. Let Pn:y1-yn≤y1-y2+y2-y3+…+yn-1-yn. Consider P2,…

Q: x + 3, x 2 x - 2

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Q: Using the definition of continuity directly prove that f : R → R defined by f(x) := x2 is continuous

A: Using the definition of continuity, the given question has been proved in step 2.

Q: Hw4 Prove if x;-→x then if xj ->x Hw4| Prove and HoS Prove if x; →× y;→y > X

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Q: 4. (i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is…

A: According to mean value theorem , If f(x)f(x) is continuous over an interval [a,b],[a,b], then there…

Q: 5 - x, -1sxS 2 r² – 1, 2<x < 3 Discuss the continuity of g(x)

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Q: Use Taylor's Theorem to find an interval [ — z, z] where the Maclaurin polynomial p2(x) - = 1 1…

A: The given function is fx=cosx and Maclaurin polynomial p2x=1-x22. The Taylor's Theorem states:  If…

Q: (1) Let f(x) = 1,x > 0 with domf= R4+. Find the conjugate of function f(x). %3D

A: given fx=1x , x&gt;0 with domf=R++ claim- find the conjugate of function fx

Q: Explain why the function has at least one zero in the given interval. C f(x) = x² - 4 - cos(x) [0,…

A: Please find attachment below. Where I explain all these things.

Q: 15. Let f : (0, 7/2) → (1, 0), f(x) = a formula for (f-')'(y). = sec x. Prove that f- is one-to-one…

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Q: Define f(x) = x² for all x. Verify the e-8 criterion for continuity at x = 2.

A: Given real valued function  f(x) = x2, ∀x We have to show that the given function is continuous at x…

Q: Let f: R/{1} → R given by f(x) ==, for every x € R/{1}. Then the given f is %3D x-1 bijective.

A: Topic:- function

Q: a) Determine the continuity of f(z) at z=2i where z² +4 f(z)={z-2i z - 2i, if z # 2i, if z = 2i.

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Q: II. Prove the following theorems on limits. 1. f(x) = L if and only if (x) – L] = 0 2. f(x) = L if…

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Q: 4. Assuming that the equation F(U, X1, X2,..., xn) = 0 implicitly defines a utility func- tian U =…

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Q: Let f : R → R be such that for all x, y ∈ R, |f(x) - f(y)| ≤ (x - y)2. Prove that f is a constant…

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Q: - Using prove M the E - f ? f(x) continuous 8 definition of continuity R > R that is defined by 3x +…

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Q: A function f:R R, where R'is the set of all real number satisfies the equati on (x) + f (y)+f(0) for…

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Q: 2. Define f : (2,7) → R by f (x) = x³ – x + 1. Use the definition of uniformly continuity to prove…

A: Consider the given information. fx=x3-x+1 As the function is in the form of a polynomial.  Now, find…

Q: 3. Let f(r) = 8 – 4x4/5. Carefully explain why f is continuous on [-1, 1), f(-1) = f(1), but there…

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.