Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then |ƒ| is also differentiable at c. (b) Give an example showing that part (a) no longer holds if f(c) = 0.

Q: В. B. Show that the function f (z) = -i z2 is analytic.

A: The solution is given as

Q: Q3-A) Is the function F(Z) continuous or not at z-U? 1+|21,

A: Let z = x+iy then Re(z) = x and |z| = √(x^2 + y^2) If z = 0 => x = 0 and y = 0

Q: 3. Find the domain of the following functions: a.) f(x) = V1-x Inx b.) f(x) = Vx - 1lr

A: solutionsince the given (3)(c)f(x)=1-xlnx(b) f(x)=x-l ln(x)use have to find the domain of the…

Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

A:

Q: Determine the function of equation (u) a (1²5) ²= if a<B? a) 2) 1 32 2 4) It is impossible Leio to…

A: Given that,  f(x) = a (x-5)2 Here a &lt; 0  Clearly  f(x) = 0 a (x-5)5 = 0 Implies (x-5)2 = 0…

Q: If y =f (x) is a function in x then prove that f(a + nh) ="C,f(a) +"C, Aƒ(a) +"C,ƒ(@) + +"C, A"…

A: Given: y=fx is a function in x Prove that: f(a+nh)=C0nfa+C1n∆fa+C2n∆2fa+…+Cnn∆nfa…

Q: (a) What does it mean if a function f: A B is not surjective?

A:

Q: 1. Describe all functions f such that f(r) ar² + bx +c and f(1) = 2 and f(-1) = 6.

A:

Q: u(1) = 2 and u' (1) -4. Find the tangent line to the curve и(х) cos(y) — 5eV — 3. at the point…

A:

Q: the following statement. 'If yes, explain why. If no, provide a counterexample. For every real…

A: If the function is injective , then the above statement is true.

Q: Let f: R2 R2 be the function given by f(x, y) = (x + y, y). Prove or →>> 8. disprove: f is…

A: Function which is one to one

Q: (c) Evaluate the Riemann integral ₁¹(x) dx by using definition with the parti- tion 2n=1,2}, n+1 =…

A:

Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

A:

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

A:

Q: ((x+1)ª-y² Consider the function g(x, y) = {(x+1)4+y²' -1, if (x, y) # (-1,0) if (x, y) = (-1,0)…

A:

Q: o f(x) f(x) is not a function because when x < 0 f(0) is undefined < 0 is undefined is defined

A: We have to complete the sentence.

Q: (f) Letf : A →→ B and g : B → C be functions. Prove that if g of is one-to-one, then f is…

A:

Q: 1; when x when x > 1. 2x + 1 1. Consider the function f: R → R defined by f(x) 5 - 3x A. Consider…

A: Given function is f(x)=2x+1,    for x≤1;5-3x,    for x&gt;1. We need to check whether the function…

Q: Consider a function f given by А f(x, y) u(x) + iv(y). Determine all possible functions f of the…

A: