5. Let f (z) be analytic in some domain. Show that f (z) is necessarily a constant if either the function f (z) is analytic or f(z) assumes only pure imaginary values in the domain.

Q: Show that the function f(x) = exp(−|x|) is not differentiable at x = 0.

A: We have given function fx=e-x As we know the modulus function is not differentiable at x=0.…

Q: Suppose that f: [0,1]–R is a function satisfying the conclusions of the intermediate value theorem.…

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Q: Examine if f(x) = ln(x2+2) is uniform continuous on the interval (-∞,∞)

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Q: Is there a real function f(x) that is differentiable only at a point x?

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Q: Suppose f(x) = - 2x - cosr. (a) Show that f is strictly decreasing on R. (b) Hence, justify that 2x…

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Q: Examine if f(x) = x3sin(1/x) is uniform continuous on the interval (0,2]

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Q: 5. Show that the function h(r, y) defined by if + y 1 is discontinuous.

A:

Q: 2.1 Use the Cauchy-Riemann equations to determine whether f(z) or not. If analytic, give the domain…

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Q: 67. Is the function given by g(x) 1 continuous over %3D x + 5 the interval (-4, 4)? Why or why not?

A: g(x)=1x+5

Q: 2.2 Use the Cauchy-Riemann equation to determine if the function f(2) =r- i(2-y)" is analytic or…

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Q: 2. f(x) = Vx – 1 at x = 5

A: Function is continuous and differentiable both at x= 5.....for detail...go through the solution

Q: if f(x) is a function that is continous at (2,f(x)) then f(x) is differentiable at (2,F(2))   true…

A: Given statement is If f(x) is a function that is continous at (2,f(x)) then f(x) is differentiable…

Q: Suppose that f (x) < 0 if x is positive and f (x) > 1 if x is negative. Can f be continuous at…

A: f(x) will be continuous at x = 0 if LHL = f(0) = RHL

Q: 5. Prove f(x) = e* + x has a root on the interval (-1,0) using the Intermediate Value Theorem.

A:

Q: What does it mean for f to be differentiable at a ?

A: we have to define that what we mean by the f is differentiable at a.

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: Show that the function is differentiable by finding values of ε1 and ε2 as designated in the…

A: Given  f(x,y)=x2y

Q: Show that f is not differentiable at x = 1 and has a corner in its graph there. x 1

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Q: Suppose f  is twice differentiable on R  and has threeroots. Show that  f " has at least one real…

A: According to Rolle's theorem, there is a zero of f'x in between two zeroes of f(x).

Q: Determine the points at which the function is not analytic: (iz^3 + 2z) / (z^2 + 1)

A:

Q: Consider the function f(z) = 2x+ 18z2-330z +37 defined on the interval -1, 10). Then the absolute…

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Q: 5. Show that the function f (x) has roots between (0,2) x² + 2x – 3 f(x) x - 1

A: To show the f(x) has roots between a and b we have to show that

Q: Show that the given function is analytic in an appropriate domain. 2- y S(x) = Tr+2}* +3y² (x+2)*…

A: Please consider the handwritten solution.

Q: Show analytically that the function f is not differentiable at x = 0. Ax) = 2x+1 x>1

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Q: Prove that constant function is differentiable at ever x ER.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: Show that f(x) = x – cos(x) has only 1 real zero on the interval [0, .

A: given fx=x-cosxx∈0,π2

Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

Q: Suppose that u(x) is a differentiable function such that

A:

Q: Determine the point at which the function is not analytic iz² – 2z ƒ(z) = 3z + 1 - i

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Q: Verify that Rolle's Theorem can be applied to the function f (z) =2³ – 10z2 +27z – 18 on the…

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Q: when that this 1x² + y² Shothathis sho (x, y) (0,0) and f(0,0) function is continous at (0,0)

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Q: Prove that f (x) = x2 takes on the value 0.5 in the interval [0, 1].

A: Here given

Q: Show that f(x)= |x| is not differentiable at x = 0.

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Q: Prove that f(x) = tan(x) is never negative in 0 < x < ; and show that g(x) = x=l sin x decreases…

A: We have to prove that f(x)=tan(x) is never negative in the interval 0,π2. Then we have to show that…

Q: 1) Are the a) f(z) = zz b) f(z) = z+z² following functions analytic?

A: Since you have asked multiple question , we will solve the first question for you. If you want any…

Q: Suppose that f hits every value between y = 0 and y = 1 on the interval [0, 1]. Must f be continuous…

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Q: Show that the function f(x) = 1 – I - x² is continuous on the interval [-1, 1].

A:

Q: 46. Let f(x) = x³ - 4x² . Find the points on the graph of f where the tangent %3D line is…

A: 46. The given function is fx=x3-4x2. Differentiate fx with respect to x. f'x=ddxx3-4x2=3x2-8x Now,…

Q: Let [x² + 1 _ if x 1 f(x) - Is f differentiable at 1?:

A:

Q: Show that the function f(z) = z = is differentiable but not analytic at z = 0

A: Givenf(z)=zz

Q: Prove: f(x)=5x y is continuous at x=2

A: A function fx is continuous at a point x=a, if limx→afx=fa Given function is

Q: 34. Show that f(x) = Jã is not differentiable at x = 0.

A:

Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.

Q: Verify Rolle's theorem for f(x) = x² - 9x +6 over the interval [0, 3].

A:

Q: B) Show that f(x) sin for x * 0. for x= 0. is not continuous.

A: