be f:   C-->C, f=u+iv an integer function. Prove that if Re(f) = u is constant. then f is constant in C. further prove that if |f|2 = u2 + v2 is constant , then f is constant

Q: Let Assume that ƒ has a continuous derivative on [a, b]. Show that there exists a polynomial p(x)…

A: Given f has continuous derivative on a,b then for all ε>0 there exist a polynomial qx such that…

Q: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

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Q: Suppose that f [a, b] → R is strictly increasing, i.e., that if x₁ < x₂ then f(x₁) < f(x₂). Explain…

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Q: Let f(x) = cos (x). %3D If it is true that f" (x) = x(f'(x))ª for some constant a, find the value of…

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Q: Show that F = ху, , zy) is not conservative. 2

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Q: Determine if f(x)=, 1+x2 is uniformly continous on the interval I = (-∞, +∞)

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Q: 6. Use an appropriate substitution to prove f(r)dr = f(a + b – x)dr. %3D

A: We have to prove definite integral property.

Q: Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

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Q: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

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Q: Find conditions on a and b that ensure f (x) = x3 + ax + b is increasing on (−∞,∞).

A: Consider the given function as fx=x3+ax+b

Q: The function f : R → R is defined by the rule f(x)=2x+3              a. Prove f(x) is one to one…

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Q: Exercise 9.23. Is there an analytic function f R→ R for which f(n) (0) = n for all n ≥ 0? If there…

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Q: if f:[0,2] 3 R such that if x>0 f(x) = { ²2 if x<0 show that whether f is continuous in [0,2]

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Q: Give an example where | f(x)dg(x) exist even through f is unbounded on [a, b].

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Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: Use Theorem 2 to show that 3, if x <-10 2, if – 10 sxso F:R → R,F(x) = -2, if 0< x< 10 -3, if x2 10…

A: so we have to show that given function is discontinuous at 2 and -2 (a)   Discontinuous  at -2…

Q: f(x) = log,(\a² x ·7*) log, (ya²x •7*).

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Q: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0.

A: Concavity of a function can be determined using the second derivative of the function. If the second…

Q: Let f(x) = x²+x+ 2 = Z₂ (x) prove that f(x) irreduciable Poly

A: The given function is  f(x)=x2+x+2.

Q: Suppose that f: R (0, ∞) and that f'(x) = f(x) ‡ 0. Prove that (f-¹)(x) = 1/x for x > 0.

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Q: If f(x) is continous on [a,b] then f(x) is differnitable on (a,b). True or False?

A: False Any differentiable function must be continuous at every point in its domain. The converse does…

Q: Determine whether or not the /VT applies to show that the givenfunction takes on all values between…

A: As we know Intermediate value theorem (IVT) states that, Suppose f(x) is continuous on [a,b] and M…

Q: Express the following functions in the form u+iv f (2) = In(2)

A: We know that the complex number z=x+iy can be expressed in the form z=r·eiθ where  r = x2+y2 and…

Q: 2. Let f: [0. ox) →R be the function given by f(r) r2 +2. Is the statement Vrvy((z) > f(y)) (x> y))…

A: We have to solve given problem:

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: Suppose f'(x) = c for all x, where c is a constant. Show that, for some constant d, f(x) = cx + d.

A: suppose f'(x) =c for all x, where c is constant .Show that for some constant d, f(x)=cx+d

Q: Let f : R → R be continuous such that f(q) = 0 for all q ∈ Q. Prove that f must be the zero function

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Q: Is the cobb-douglas function strictly concave when a1 +a2 <1 when

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Q: Suppose that f(x) = SBx? – 3 if if (Bx+2 x> 2 Find the value of B so that f(x) is continuous…

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Q: Use the MVT to prove that if f' (x) ≤ 2 for x &gt; 0 and f (0) = 4, then f (x) ≤ 2x + 4 for all x ≥…

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Q: Assume that f(z) = ez then Re(f(z)) and Im (f(z))are

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Q: Let f(æ) = (4x + 26) In(æ + 3) %3D for æ > -3. Find the smallest value of A so that f(a) is concave…

A: Here we will find out the required condition.

Q: Use the definition to prove that the function in f(x) = 2x - 3 is uniformly continuous.

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Q: Compute P(0), and find the exact values of the six circular functions given that 0 - 135°. = -

A: θ=-135° Now x-coordinate of Pθ is, cos-135°=cos135°   ∵cos-x=cosx=cos180°-45°=-cos45°…

Q: If f is increasing on the interval [a, b), then f' does exist. Select one: a. True b. False

A: Let f is increasing on the interval [a, b]. Since, there is not given that the function is…

Q: Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does…

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Q: Define f: [0,2]-R by f(x) = x²+2x-1 , then the sup f- inf f is : [0,2] [0,2] .a 3 .b 8 .C 8, .d 1

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Q: Let f be a function defined by: f(x) = ln(22x). In Then, f-¹(x) = e¯* + 2. ос إختر واحداً: True…

A: Topic = Functions TRUE

Q: Suppose that f(x) is continuous and differentiable, f(2) = -2, f(6) = -6, and f(x)dx =-10. Compute ,…

A: Topic:- Functions

Q: Find the interval(s) on which f(x) = 2x^3-3x^2-12x+1 is concave up.

A: Given : fx = 2x3 - 3x2 - 12x + 1

Q: Determine if f(x)=In (x² + 1) is uniformly continous on the interval I (-0, +0)

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Q: Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.…

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Q: Assume that f'' exists and f'' (x) = 0 for all x. Prove that f (x) = mx + b, where m = f'(0) and b =…

A: Given that  f'' exists and f'' (x) = 0 for all x. Integrating both sides ,we get By substituting…

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: 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: 1. If f(x) = x + v2 – x and g(u) = u + /2 - u, is it true that f = g?

A: Since you have asked  multiple questions in single request  so we  will be answering only first…

Q: Suppose that x = x(t) and y = y(t) are both functions of t. If 22 + y = 20, and -1 when x = 2 and y…

A: x2 + y2 = 20 Differentiate with respect to t 2x dx/dt + 2y dy/dt = 0 In this equation we put x = 2,…

Q: Let f : 0, 1] R be defined by f(x) = {1+a, a € Q .2 Then Jo f is

A: Given: f:0,1→ℝ defined by fx=1+x, x∈ℚx+x2, x∉ℚ To find: ∫01f=? Used formulas, ∫xdx=xn+1n+1∫dx=x