Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

Q: Define an inner product space for continuous functions on [0, 1] as (S, g) = [f(x)g(x)ax with the…

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Q: (a) Assume that g(x, y) is a function of the form x²+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that…

A: (a) The given function is gx,y=x3yx2+y2  if x,y≠0,00             if x,y=0,0. To show: (i) gx,y≤xy…

Q: Examine the continuity of f(z) = 2-√z+4 Z at z = 0

A: Given: fz=2-z+4z We have to find the continuity of f(z) at z=0.

Q: Show that each function in satisfies a Laplaceequation. ƒ(x, y, z) = 2z3 - 3(x2 + y2)z

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Q: Let f:R→R satisfy f(x+y) =f(x) +f(y), ∀x,y∈R. If limx→0f(x) exists, prove that f is continuous on R

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Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Find the divergence of the field. F=(-x+8y - 3z)i+ (8x- 4y + 3z)j+ (- 4x- y- 3z)k

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Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: Find the Wronskian for the set of functions.{x, e−x, ex}

A: Find the Wronskian for the set of functions.

Q: Let f be a real valued function defined on [a, b] . Let f have a local minimum at a point x ∈ (a, b)…

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Q: Let F = <5x^2, 2(x+y)^2, 9(x+y+z)^2> Find divF (divergence)

A: First of all, consider the vector F and find F1, F2, and F3.

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

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Q: Find the Wronskian for the set of functions.{1, x, ex}

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Q: Let f be a function defined on R satisfying |f(x) – f(y)| < ÷ læ – yl, Væ, y E R. a. Show that f is…

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Q: Let f : R → R be a function such that |f (x) – f(y)| 0. Write the e-d definition of continuous…

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Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

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Q: Let f:X → Y be function and g: X → fcX ×Y, g(x) = (x, f(x)). Prove that if g is a homeomorphism then…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)Df(x)+fy) vx, yER. Show that…

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Q: Let f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

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Q: Find the local linearization of f(z) =z² at 9. lo(z) = %3D

A: Given : f(x) = x2   x0 =  9

Q: (a) Assume that g(x, y) is a function of the form g(x, y) = {" 0 if (x, y) = (0,0). (i) Show that…

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Q: Find the linearization L(x) of f(x) at x=a f(x)= x+1/x a=3 show all work

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Q: Let f: X_nte, Y be X,Y are Tj spaces. show that: Xö locally compact iff Y is so . perfect function,…

A: Given: f:X→ontoY is a perfect function, where X,Y are T2-spaces. To show that X is locally compact…

Q: Let H(x) integral ,for -2 ≤ x ≤2.Use geometry to evaluate H(2).

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: Let f,(x) = x E R. Prove that f 0 uniformly on R. COs nr %3D

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Q: Let f be a function defined on R satisfying If(z) - f(y)| < 히 - J), V,yER. a. Show that f is a…

A: Given that f be a function defined on ℝ satisfying…

Q: find the divergence of the field. 1. F = (x - y + z)i + (2x + y - z)j + (3x + 2y - 2z)k 2. F = (x ln…

A: Consider the vector field Substitute the value of vector fields then Hence, the divergence of…

Q: Use the definition of uniform continuity (i.e., the e, d definition) to prove that f(x) = x² – x is…

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Q: Show that the function f defined by 1, (x, y) = (1,–1) f(r, y) = { x² + y (x, y) # (1,–1) r+ y is…

A: In this question, the concept of limit is applied. Limit Limits are the points at which a function…

Q: Consider a surjective function f : A → A Prove that fo f o f is surjective

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Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by 2.x2 f…

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e"…

A: Differentiate the equation wrt x:  ddxyx=ddxex+yx22y'x=ddxex+ddxyx22y'x=ex+xy'x2

Q: Let F be defined on R x R to R by F(x, y) = x² – y. Show that F is C' on a neighborhood of (0, 0)…

A: Let F be defined on RXR→R by F(x,y) =x2-y. Now we have to show that F is in class C' on a…

Q: Show that the function f defined by 1, f(r, y) = { r2 + y (x, y) = (1, –1) (x, y) # (1, –1) is not…

A: We have to show that  f(x, y)  is not continuous at  (1, -1)

Q: Prove that for any Riemann integrable function f on the interval [a, b] there is c such that f = [°…

A: let f be Riemann integral an [a, b] then clearly f is bounded.

Q: b) Use the divergence theorem to compute f. F ·ñ dS, where S is bounded by x + y + 2z = 2 located in…

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Q: Find and describe the points in the space that the function f(x, y,z)=In(xyz) is continuous.

A: We have to find and describe the points in the space that the function f(x,y,z)=ln(xyz) is…

Q: Let f : (0, 1) → R given by f(x) = 1/x . (i) Prove that f is continuous. (ii) Prove that f is not…

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Q: 3. Evaluate the limit r(t +h) - r(t) lim h→0 h for r(t) = (t-1, sin(t), 4).

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Q: Let f : [a, b] → R be a strictly decreasing function. Prove that f is integrable [a, b). on

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Q: Let Juce) 2ntv () %3D A=O n!F (I+u+n) Bessel function Of the first Kind be a Of Order V Show that…

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Q: Let F(x, y, z) = 4z²ei+ (y³ + tan(z))j+ (4x²z – 3y²)k. Use the Divergence Theorem to evaluate f, F .…

A: Given,       F(x, y, z)=4z2xi+43y3+tan(z)j+(4x2z-3y2)k use the divergence theorem to evaluate…

Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

Q: Let f be an integrable function of [a,b] and define F(x): = | f for x E [a,b]; 1. Evaluate G(x): = |…

A: Let f be an integrable function of a,b and define Fx=∫axf for x∈a,b Note that ∫axf=∫acf+∫cxf for…

Q: Let f,g: D -> R be conitnuous at c ∈ D. Prove that fg is continuous at c.

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Q: ) Denote the number of derangements of {1, 2,..., n} by Dn. Using the formula for the number of the…

A: Given Dn be number of derangements of 1,2,…,n. Derangements is number of permutation of n objects in…

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.