3. Suppose that ƒ € C'(B1,Rk) satisfies ||Df(0)|| = }. Prove that 3 6 > 0 such that %3D |f(1x) – f(y)| < |T – y| Vx, y E Bs.

Q: 1 Let f (x,y) =In Then f (1, – 1) =... 3x2+4y3 XX

A:

Q: Check it for f(x, y, z) = x²y°z*, that is, compute Vf first, then V x (Vf).

A:

Q: 2. Show that the following functions are convex by verifying the condition that V²f(x) 20 is…

A:

Q: 1. Let F(x, y) = 25x³ + 120x²y + 192xy? + 102y³ – 300x – 450y. Find the following: | ƏF (i). OF…

A:

Q: Let f(x) be a continuous function satisfying f(x) = (x² + 1) || f(t) dt + 2x. Compute f(0.9)…

A:

Q: 46. Prove that there is no continuous function f: R→ R such that, for each c E R, the equation f(x)…

A: Suppose such continuous function  f  exists.  For each c in R, the equation f(x)=c has exactly two…

Q: 3. Check that in each of the following cases, the equation F = 0 defines y locally as a C' function…

A:

Q: 7. Prove that (a) f (x) = / eika F (k) dk = F-1{F(k)}, %3D %3D = deikb/a F (k/a). 1 (b) F [f (ax –…

A:

Q: f:N → N,Vx E N, f(X) = 5x² +3 g:N → N,Vx EN, g(x) = 3x – 2 (gof)(x) find (1

A:

Q: Suppose that P1(t), p2(t), and p3(t) are continuous on R, and that y1, Y2, and y3 are solutions to…

A: Wronskian of the function

Q: b) Let X = Cle,e+ 2] and define T:C[e,e+2] → R by re+2 T(z) = | 1(t)dt, r€ Cle,e+2] Show that T is…

A:

Q: Let f, g: R → R defined by if x 1 f(x) = %3D and g(x) = 3x + 2. %3D fog and go f.

A:

Q: Let F(z, v)-v +y+3z+1-0

A:

Q: Prove that = eika F (k) dk = F-1 {F (k)} 1 (a) ƒ (x) = (b) F [f (ax – b)] = eikb/a F (k/a).

A: 7. The problems related to Fourier transform are solved as shown below.

Q: Consider f e C} (R") and the following function 1 Hy (æ, z) = f(z) + Vf(z)"(x – z) + ||x – z||3, 27…

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

Q: 7.2 Give an & - 8 pr at (1 continuous (2x - 2).

A:

Q: 5. Prove that |xy| = |x|-|y| for x, y eR.

A:

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

A:

Q: 6. Show that f(x) = sin x is uniformly continous on R.

A:

Q: Prove that f (z) =zz is nowhere analytic

A:

Q: 24. If f(x, y)=x²y+cosy+ ysinx, find fu, fyy & fry.

A:

Q: 4. Prove that the function f: [0, ∞)→ R; f(x) continuous. = √ is uniformly

A:

Q: 1. Let f : (−1,1) → R, f(x) = (1+x³)-³. Compute g' : (0, 1) → R where (a) g(x) = X X f. (b) g(x) =…

A: Answers along with explanations are provided in the image. Explanation:

Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

A:

Q: Show that a continuous function f : [0, 1] → Z is a constant function.

A: Consider a function .We need to show that a continuous function  is a constant function.We know…

Q: 3. Find and sketch the domains of f(x, y) = xy and g(x, y) = xy In(9-x² - 9y²)*

A: The given functions are,

Q: Use the equation ƒ(x) = ƒ(a) + ƒ′(a)(x - a) + ƒ′′(c2) /2 (x - a)2 to establish the following test.…

A: For the given function to determine its behaviour at a critical point x=a, use second derivative…

Q: F . dr, (c) Let f(x, y, z) = x³ + 2yª + 325 and F consists of three line segments from (1,0, 0) to…

A:

Q: Let f(x, y) = 6e sin(7y) af af dy

A:

Q: 1. Show that if f: [a,b] → R and a < xo <b, then f'(xo) exists if and only if D f(x) = Dƒ(x) € R.

A:

Q: 1. Let f(u, v) be a continuously differentiable function. Let u = e*, v = e³. Show that u²fuu + v²…

A: Given that f(u,v) is a continuously differentiable function, and u=ex, v=ey then show that…

Q: Suppose f: -1, 1] → R is continuous and satisfies f(-1) = f(1). Prove that there exists 7 € [0, 1]…

A: Given is a continuous function.

Q: 1. For T € L(V,W), prove that null(T) = (range(T*)) and range(T) = (null(T*))+.

A: Given these, we can now prove the statements:  Proving (null(T)=(range(T∗))⊥):By definition, for any…

Q: 3. Let fx) = {ax), whire a function. sadafferentiable 10 94x) 1t xa aa glx) and a'cx) Lalwlate the…

A:

Q: f(x + a) dx = f(1) dæ = | f(x) d.r. -T+a

A: Proving of integral of periodic function

Q: are defined c chy's mean

A: