Suppose : R →R is a convex function with /(0) = 0. Show that, for all a, y E dom f and for all 01,02 € [0, ] such that 1+02 = f(6, + 624) < O1f(#) + 02f(y).

Q: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

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Q: Please

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Q: Consider the regular subdivsIon of the interval [a, b] as a = x0 < x1 < x2 < x3 < x4 = b, with the…

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Q: Show that the limit of the function f(x,y)= (0,0). (x+y)² x² + y² is indeterminate at

A: Limit of the function

Q: Let g=yf(xy). Show that: - дд у ду дд - Х = g дх -

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Q: Is the following bounded or unbounded and why? f(x,y) = ln(x2 + y2)

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Q: If (Xo, ..., XN) is a martingale and the function f : R → R is convex, then (f(X,),..., f(XN)) is a…

A: Please see the attachment According to guidline we are supposed to do one problem.

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

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Q: (b) Show that the u(x, t) can be written as u(x, t) = 0(x – at) + 4(x + at), where o and are…

A: The multivariable function u(x,t) is a rule that associates or fixes a number to a particular point…

Q: Prove the following statement: For any function f : R" - R and any function g : R" R, let h: R" R be…

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Q: Show that the functionf (x ,y)=xy³onD: x|<2, ]y|<oodoes not satisfies Lipschitz condition.

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Q: Show that f(x) = In x is not uniformly continuous on (0,2)

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Q: If after evaluating the Wronskian of the functions y1 and y2 the result is W=7x2+19 in the interval…

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Q: The Sandwich Theorem for functions of two variables states that if 8 (x, y)s f(x, y)sh(x,y) for all…

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Q: Analyze the following function for the necessary and sufficient conditions of Schwarz's theorem. ху…

A: f(x,y)=xyx2+y2 fx(x,y) =x2+y2 y -xy2xx2+y2 (x2+y2 )2           =x2y+y2y -2x2y(x2+y2 )3…

Q: Suppose that f(x, y) is a differentiable function of two variables defined on the entire xy–plane,…

A: Let us consider a function f:ℝ2→ℝ defined by  fx,y=x2y2                                      (i)…

Q: The domain of the function f (x, y, z) = In(3ry – 6) is %3D - O {(x, y) E R² : xy 2} o {(z, y, z) E…

A: Given function is: f(x,y,z) = ln(3xy-6) We need to find the domain  for this function.This can be…

Q: If R is a closed region on the xy-plane andf is a continuous function on R, then f has a relative…

A: The given statement is True.

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

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Q: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

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Q: a. We say that a function ƒ : [a,b] that f(x) 0 > Df(x). R has a local minimum at x if there exist…

A: a. Given that a function f : a, b→ℝ has a local minimum at x if there exist h&gt;0 such that…

Q: Show that the limit of the function f(x,y) = at (0,0) does not exist. (x+y)² x² + y²

A: Limit of a Function: Like a single-variable function, a multi-variable function can also have limit.…

Q: Suppose f'(x0) = 0, f"(xo) = 0, ..., f(n-1)(x0) = 0 and f(") (xo) > 0, for a C function f. Prove…

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Q: Suppose that a continuous function g satisfies: | = 36 and g(x)dx | 9(2x)dr = 15. 0. : = =p(= |…

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Q: 4. If L is a Continuous function on H, then unique YEH such that L(x)=(x,y). уєн there is a

A: Given that L is continuous on H. To show: y is unique in H.

Q: (a) Prove that if m ₹ N, then the function f(x) = xm is continuous on R.

A: We have to prove that, the function f(x) = xm,   m∈ℕ  is continuous on ℝ.

Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

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Q: Find aland U so that the function S 7z³ – 3x² + 7, z -2 is both continuous and differentiable. al =

A: To Determine: fins a and b so that the function is continuous and differentiable given: we have a…

Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

Q: Are the following sets of functions, each defined on x € R, linearly independent? Use Wronskian. (a)…

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Q: Let A C Rª be a convex set. f : A → R is a strictly quasi-concave function if and only if for any a…

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Q: Let f be defined on [a, b], if f has a local maximum at a point x E (a, b), and if f'(x) exists,…

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Q: Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

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Q: Suppose f is differentiable on (a, b). If c E (a, b) and f' (c) = 0, thenf (c) is either the maximum…

A: Given f is differentiable on (a,b). and c lies in (a,b) such that f'(c)=0.

Q: Let f = f(x, y), x = s +t and y = s - t. Show that (9- ) - se se as at ax ду

A: It is given that, f=f(x,y), x=s+t and y=s-t. Show that, ∂f∂x2−∂f∂y2=∂f∂s·∂f∂t.…

Q: 4|x| – 3 pour x 1, x² – 4 f2(x) + V2 + x², || ||

A: Given that, f1x=4x-3x≤1-3xx2-4x&gt;1f2x=1+2+x2 To determine set of points which they are derivable…

Q: The domain of the function f (x, y, z) = In(6 – 3ry) is - o {(x, y) E R² : xy > 2} O {(x, y, z) E…

A: Please refer to the image below

Q: If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly…

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Q: Show that f (x, y) = has no limit as (x, y)→(0,0) Vx2+y2

A: The objective is to show that f(x,y) has no limit as (x,y) approaches to (0,0).

Q: Let RC [0, 1] ×x [0, 1] such that (x, y) E R (i.e., xRy) whenever 4x < 6y < 9x . Deter- mine if R is…

A: R is not a function, because there can be multiple ordered pairs with the same first coordinate,…

Q: Suppose f'(x) is continuous over an interval a <¤< b, and a <c<d <b. If f'(c) = 8, then (check all…

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Q: The largest region, on which f(x,y,z)= 2-, is continuous, is y+1 x²+z²-2 O All points not on the…

A: According to question given thatf(x,y,z)=y+1x2+z2-2The largest region on which f(x,y,z) is…

Q: Suppose that f(x)=xu(x), where u(x) is a differentiable function at x=2 such that u(2)=3, u'(2)=1.…

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Q: Assume that is a continuous real function on (a, b) such that 1 1 o ( x + ³) ≤ ²}/20 (x) + = 0 ( for…

A: Here given that φ is a continuous real function on a, b such that φx+y2≤12φx+12φx

Q: Let f(x) = x² In x, where x > 0. Show that f(x) has only one critical point.

A: Given data: The given function is f(x)=x2lnx. The critical point is the point where the slope of the…

Q: Suppose that 36f'<x) <5 for all valnes of X. What are the minimum and maximum possiable Valises of…

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Q: Analyze the following function for the necessary and sufficient conditions of Schwarz's theorem. (x,…

A: Schwarz's theorem states that for a function f:D→ℝ is defined on set D⊂ℝn, if (x0,y0) is a point…

Q: Find the Wronskian for the set of functions.{x, x2, ex, e−x}

A: Given information: x, x2, ex, e-x The Wronskian is given by the following determinant.…

Q: 5) Suppose a path C is defined by r(t), 2 and r(8) = . If f is a differentiable function satisfying…

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Q: Show that if u = x – y and v = x + y, then for any continuous f, -2sx * f(x – y, x + y)dydx = / |…

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