xMyn x2+y2 (х, у) 7 (0, 0) f(x, у) %D (х, у) %3D (0, 0)

Q: 2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor…

A: Given the function f:ℝ2→ℝ defined by fx,y=1-x2-4y2

Q: 7- The function f(z) = z is: a) Analytic on C b) Analytic only at zo = 0 c) Not analytic

A:

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

A: Given f (x) = px2+2qx +r & p>0Since f(x) is a quadritic function , for f(x) ≥0,…

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A:

Q: 25-let F(xs.2)=x-x22 @ Find VF 6 write the equution of tngent Plame to the at Paint (2,1,1). Find…

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Q: Find the critical numbers of the function f(z) = z2 - z / Answer (separate by commas): z =

A: Solve the given problem.

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: If f : (0, 1] → R is defined as if z is rational S(x) = I ifr is irrational. cuss the Riemann…

A: Here the f is bounded on [0, 1]. Let Pn be the partition of [0, 1] defined by Pn = (x0 , x1, x2, ……

Q: Suppose f, g:C → C are differentiable. Prove that fg is differentiable and that (fg)' = f'g + fg'.

A:

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

A:

Q: 3 Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0 < |e'(x)| < |f'(x)| for all x €…

A:

Q: Suppose f(z) = u(x, y) + iv(x, y), and suppose that u(r, y) = x2 - y².What should v be in order for…

A:

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

A:

Q: Consider the cubic functionf (x) = ax3 + bx2 + cx + d, where a  0. Show that f canhave zero, one,…

A: We have fx=ax3+bx2+cx+d This implies  f'x=3ax2+2bx+c Set f'x=0 i.e 3ax2+2bx+c=0 and solve it using…

Q: 7. Find a and b such that f is differentiable everywhere. f(x) = {+b, S ax³, lx² + b, x 4

A:

Q: Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

A: f is increasing if first derivative is positive in the interval  and f has concave if second…

Q: Let ƒ : f: R” → R be a function given by ƒ(x1,x2,...,xn) = x².x² …x², where n x² = 1. Show that the…

A:

Q: Q4. Prove that f(z) = Iz/2 never be holomorphic function at z = 0. %3D

A:

Q: Use the Cauchy-Riemann Equations to show that f(z) = z is differentiable for 16. all z and find…

A:

Q: Consider Utt= 4UTT u(x, 0) = u(x), u₁(x, 0) = u₁(x), on the domain -∞<x<∞, 0≤t<∞. Suppose that both…

A: Given: Let us consider the given data, utt=4uxxux,0=u0xutx,0=u1x The domain -∞&lt;x&lt;∞, 0≤t≤∞.…

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

A:

Q: Use Taylor's Theorem to find an interval [−z,z][-z,z] where the Maclaurin polynomial p3(x)=x−x^3/6…

A: Taylor's Theorem: Let f has a Taylor polynomial of degree n expansion around x=c, then the remainder…

Q: Suppose f(z) is analytic for |z| &lt;= 1, and |f(z)| &gt;=1. If f(0) =1, prove that f(z) is…

A: To prove that the given conditions on the analytic function force it to be the constant function…

Q: Suppose f is known to be differentiable and contains a local extremum at c, then which of the…

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Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

A:

Q: (b) For f(z) find the residue of f (z) at the singular point (z+1)2(z2+4) z = -1

A:

Q: 1. Determine whether k(x) is continuous at r = 3, where %3D x + 3, x 2.

A: Given  k(x)=x+3,  x&lt;0x2+1, 0≤x≤22x,   x&gt;2 To check the continuity we can use the graph as…

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: Determine the value of m such that the function 5 + 5 y fi^ - ^ - f(z, y) = is continuous at (0,0).

A: Given that function s continuous at (0,0). We need to find the value of m.

Q: (5) Prove that if f is continuously differentiable on [a, b], then |f(a) – f(b)| a – b max f'(x)|…

A:

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: Defn `x;>-` "x; diverges to minus infinity" VR>O such that Vj >Ng *;<-R NR HWs If x→0 then → O.…

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Q: O Show that the function f: N2 N defined by f(x, y) = 2X3Y is injective but not surjective.

A:

Q: If the functions f(x², y²) and g(x, y) are homogeneous of degree 2 and 4, f (x, y) g(x, y) 1…

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Q: Use the Intermediate Value Theorem to show there is a root of the given equation in the specified…

A: Intermediate value theorem states that if “f” be a continuous function over a closed interval [a, b]…

Q: 3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) =…

A: Given:- f: R2→R

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.…

A:

Q: Let S«) = {r²+2x}°, 121, ax+bx, x< 1. Compute for all a and b such that the function f(x) is…

A: We have to find

Q: yLet f(2) be analytic in {z: |2| < 1} with f(0) that |f"(0)| < 2. = 0 and |f(2)| < 1 for all z in…

A:

Q: Suppose that f:I → R and g:I R are differentiable at ce I. Prove that if k E R, then the function kf…

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Q: Prove that the extreme value theorem doesn’t work over ℚ. That is, find a continuous function, f: ℚ…

A: Consider the function f: ℚ→ℚ, where f(x)=sin(x). Consider the function in the closed interval 0, 2…

Q: Show that there is no differentiable function f on [0, 2] with f (0) = -4, f (2) = 4 and %3D f' (x)…

A: Given: A function f such that f(0)=-4,f(2)=4 and f'(x)≤3, ∀x∈[0,2] According to Mean Value Theorem,…

Q: Assume thatf is differentiable in (-oo, x) and af'(r)N 0 for all a E (-00, 00). Prove that f has the…

A:

Q: 5.11. Prove that the function f(z) = Arg(z) is discontinuous at each %3D point on the nonpositive…

A: Solution

Q: 9. Show that if the analytic function f(z) has a zero of order N at zo, then f(z) satisfying g'(zo)…

A:

Q: 2. Show that the function. 5x²³²Bin 12 F(x)= โ x2 0 (x=0) is not of bounded variation on [1,1].…

A: