Let A = [0, 1] × [0, 1] C R². Suppose

Q: Given f(x,y, z) = In(x² + y² + z²), show that ² f , ²f , ² f x² + y2 + z2

A: Given problem:-

Q: (b) En=1 m

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Q: Let x, y, z E R. Prove that if x – 2y + 3z is odd, then 4xy – 6z is even.

A:

Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: Give the condition which ensure that Je-| < 1 where z E C.

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Q: Prove that r(z) = (In-)²-'dt , Re(z) > 0.

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Q: - Let X1, X2, ...,Xn~N(0,0) and T = Fi (T),Var(T)

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Q: Given u(x, t) = t-²e¯, 2 , compute:

A: We can find the required answer as below.

Q: Let X~U (0,0), then P(X > X²) is

A: Given that X~U0,θ fx=1θ-0=1θ, 0≤X≤θ

Q: Suppose that x̅ = 12, Sx = 2, and z is 1. what is X?

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Q: s(2) f(z) is analytic in C

A: Ans : false

Q: Prove that c = cn. i=1

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Q: Show and use d(Xn, FCN))<« d(Xn-1 'x) Vny? To prove X = fx) %3D

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Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

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Q: . Show that z)= E kz is D: 1215 continuous io domain

A: The series can be shown as continuous in a domain if it converges. Therefore, the first calculate…

Q: Let X and Y be defined as follows: X = {a E Z | a = 3x for some x E Z}, and Y = {b E Z |b = 3y – 3…

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Q: Let X ~ N(0, 1). For any € > 0, show that P(|X| > c) ≤ ²c - ??

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: J prove that for 0 <r<R

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Q: Prove that TIN) F(a – p) O< Re

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Q: prove that erf (x) dx =o

A: To prove: ∫−aaerfxdx=0

Q: Suppose that C> 1. Prove that if t > Vm ln C+In C, then ()/F) >C.

A: We will use the following Strining's approximation :  n! ≈ √(2πn) e-n nn To prove the given…

Q: Let x e R,-1. Prove that VNEN (1+x)" >1+ nx

A:

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

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Q: Let Z ~ N(0,1). Prove for all z 2 0, P(Z > x) S

A: It is an important part of statistics. It is widely used.

Q: If Σa, is divergent and Eb, is divergent, then Z(a + b) is divergent.

A: The given statement is false as sum of two divergent series may converge. We illustrate this as…

Q: Let f(x, y, z) = x³ – yz?. Find af af of and ax' dy az

A: Given : fx,y,z = x3-yz2

Q: nd C such that AC=E

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Q: ronskian of the DE : xp(x)y' + 3y = 0 = er² et? et?

A:

Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: 28. (a) Prove that for x e R" and s < t, Ds(x) C D;(x).

A: The notation Dsx represent  an open ball of radius s, centered at x∈ℝn. Let x=x1, x2, …, xn,…

Q: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: Let f(r, y) = In (z³y) (1,3) = 3. then Select one: O True O False

A:

Q: Let BE Cnxn. If z" Bz = 0 for all z E C, then show that B = 0.

A: Given expression is

Q: Show that if X gamma(r, A) and Y ~ gamma(s, ) independent, are then X + Y is gamma(r + s, ).

A: X~Gammar,λ fx=λrΓre-λxxr-1,     x&gt;0, r&gt;0, λ&gt;0 Y~Gammas,λ fy=λsΓse-λyys-1,     y&gt;0,…

Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

A:

Q: Let P= {p:p=6x-1, for some x EZ} and Q= {q:q=6y+5, for some y €Z}. Prove that P= Q

A: Answer

Q: Let 0 E A C R. Show that if A is both open and closed, then A =R.

A: Given that Let 0∈A⊂R.Show that if A is both open and closed, then A=R. Here we use the principle of…

Q: Prove that (A = B) where (A = ax + ay + az) and (B = bx + by + bz)?

A: Given- A=ax+ay+az and B=bx+by+bz To Prove- That A=B

Q: Let r E Rt. Prove that if z – 2x2 +2 2021. -

A: Let x2=t, Now if x∈R+⇒t∈R+ ⇒t2-2t+2≤0⇒t-12+1≤0⇒t-12≤-1which is not possible for any value of t, for…

Q: Given the function z = In (2xy + e3x) - 5y, %3D az a. ax

A: Since, we have to find the partial derivatives ∂z∂x and ∂z∂y of functiongiven by z=ln…

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

A:

Q: I c exists in (0, 1) such that f(c) (1)- f(0) 1-0

A:

Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: (a) Prove that if f e C (R²) then a (af af %3D

A: This is a problem of the calculus of several variables, higher dimensional calculus.

Q: Show that Oa = {a} if and only if a e Z(G).

A: For a∈G, Oa denote the orbit of 'a' under the conjuction action.The orbits under the conjucation…

Q: Let A + 0 and ACB. Show that NB CNA.

A: Given :A≠∅A ⊆ B

Q: Prove that (A')' = A for each A E Mmxn(F).

A:

Q: Let P= (p:p=6x-1, for some x EZ) and Q = {q:q=6y+5, for some y EZ}. Prove that P=Q

A: