Suppose that f e H(C) and |f(2)I < eRez for all z. Show that f(z) ce for some constant c.

Q: f is continuous iff f-1(E) is closed

A:

Q: Find c such that the function 10 г c - || 26 х > с is continuous everywhere. c =

A:

Q: True/False: If ƒ is entire and |f(z)| > 2 for all z E C with |2| > 1, then f is a constant function.

A: If f is entire and fz≥2 for all z∈ℂ with z≥1Let gz=1fzas fz≠0 for any z∈ℂ⇒g is entire gz=1fz&lt;12…

Q: 1. Show that for all f e L'(R) lim dr = 0. %3D

A:

Q: If f(0) = 0 and f"(x) exists for all x>0, then show that %3D f(x) f'(x) xf"(5), 0 < § < x. Also…

A: Let's find.

Q: Find the value of the constant k so that f(x) is continuous for all real values of x where f(x) is…

A: Solution

Q: Suppose that f(x) is differentiable function such that f(1)=3 and f(1)=4. If g(x)= f(x) -, then what…

A:

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: Find a constant c such that x² – c2 if x 4 g(x) ={ is continuous everywhere.

A:

Q: if r<3 if x23 -m 5) Find m so that g(x) %3D mx Is continuous for all x?

A:

Q: Find c so that f is continuous at every point f(x) = 5x+12c, if x < -5 f(x) = (cx^2)+1, if x ≥ -5

A: Given- fx=5x+12c if x&lt;-5cx2+1   if x≥-5 To find- The value of c that  the above function is…

Q: If f(x) is differentiable and strictly increasing on R, then f'(x) > 0 for all x.

A:

Q: Find all a E Z such that f(x) =

A:

Q: Show that f (x) = 1 − |x| satisfies the conclusion of the MVT on [a, b] if both a and b are positive…

A:

Q: Suppose that C is a constant. Let Cr +1 f(x) = 7 +I+1° - , x E R.

A:

Q: If G(x,y) is differentiable with respect to x and y then it is always possible to find the number…

A:

Q: Find all values of "c" such that the function f(x) is continuous over  ( − ∞ , ∞ ) f(x) = 1 - x^2…

A:

Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

A:

Q: Let f: R R and let f(x) >0 for all xER. Show that f(x) is strictly increasing if and only if the…

A:

Q: 2. Let f(x) = e" + for x E [0,3]. Determine the length of the graph of f. 4e

A:

Q: Sßx² – 3 (Вх + 2 if (b) Suppose that f(x) if Find the value of ß so that f(x) is continuous…

A:

Q: Suppose that the function f: [0, 1]  in R is continuous and thatf (x)&gt;= 2     if 0 &lt;= x &lt;…

A:

Q: Given a set E c (0,1) of measure 0, construct a continuous non-decreasing function f : [0, 1] → R…

A: We will use knowledge of measure theory and real analysis to answer this question. As instructed in…

Q: , for x 1 Let f(x) be a function de fined by f(x)3= Discuss the contuniuty of f andf".

A:

Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

A:

Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

A:

Q: Suppose that f: R → R and g : R → R are related by the formula f (x) = g(x)+(x-a)". For which values…

A: Note: A function fx is tangent to the function gx at a point x=a if (1) fa=ga, and (2) f'a=g'a

Q: For f (x) = , given e < 1, what value of d proves the continuity of f at x = 1? (You may assume that…

A:

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: 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: If we apply Rolle's Theorem to the function f(x) = 2x- 16x 6 on the interval [2, 6|, how many values…

A:

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

A:

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: Suppose f is entire and f(2)| s |2/0.9 for all z E C. Prove thatf is identically 0. (Hint: Follow…

A:

Q: Given that f(z) = z where does f'(z) exist

A: The given function is,  f(z) = z. We have to find those points where f'(z) exists.

Q: Suppose that f : R → [0, 0) is measurable. (positive) show that if f R then F=0 a.e

A: For the solution follow the next step.

Q: For f(x) = x - 2x + 1, find all the values of c on the interval [0, 2] that satisfy the Mean Value…

A: Follow the steps.

Q: Let F(x) = Vg(x). Find a formula for F'(x) in terms of g(x) and g'(x). Use your formula from (a) and…

A: Fx=gx Differentiating with respect to x, we get F/x=12gx·dgxdx    ∵dxdx=12x=12gx·g/x=g/x2gx

Q: If f"(c) = 0, then f(x) cannot have a local extremum at x = c.

A: We know, if f"(c) is negative then f(x) have local maximum at x=c and if f"(c) is positive then f(x)…

Q: (b) Suppose that f(x) = ൜βx2 − 3    if x ≤ 2βx + 2      if x &gt; 2 Find the value of β so that f(x)…

A: For continuity, left hand limit=right hand limit=f(x)

Q: For f(x) = {²-1 if > 2 5-x if x < 2 Of is not continuous at x = 2 Of is differentiable at x = = 2 Of…

A:

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: (e) If f(x) is continuous at x = 0, then what is the value of n: if 20 if x=0 f(x)=\ 38

A:

Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

A:

Q: Then the value of k for which f(x) is continuous for all real values of r is

A:

Q: If f(x) = then f (x) at z = 1 is

A: Given that :-         f(x)=1x         f'(x)=? at x=1

Q: (b) Suppose that f(x) =  βx2 − 3            if x ≤ 2 βx + 2             if x > 2 Find the value of β…

A:

Q: Find e so that f(x) is continuous on the entire number line. if x3

A: fx=x2    if  x≤3cx    if  x&gt;3 Since the given function is a polynomial function for all x&lt;3…

Q: Find c such that the function x c is continuous everywhere. x² – 8 f(x) = {

A: ANSWER : c = 5

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.