Given that f : a In (1+ x) for x E R, r > -1, then f: x → O A. (1+2) O B. In (1+2) 1 ос. е - 1 O D. (--1) O E. - In(1+ a)

Q: What is the domain of the following function? f(z,9) 4-2 - y - In(y - z) + sin(z) +y 24 and y> =)…

A: Domain.

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: 55 Use abs(x) for |x|. Given that f(x) = |x| and g(x) = 5x + 5, calculate %3D 6. (a) ƒ o g(x) = %3D…

A: Given, f(x) = |x| and g(x) = 5x+5 (a) f o g(x) = f(g(x)) = |5x+5| (b) g o f(x) = g(f(x)) = 5|x| + 5…

Q: Prove that (r) = tan(r) is uever negative in 0 <r< and show that g(x) = xsinr decreases…

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

Q: Define fi z-> 2 xZ by Ala)l (2,72t1) if xis even 18,0) if t is odd a) Is f injective 2? b) Is f…

A: We have f:Z→z×z Such that fx=x,x+1     if x is evenx,0          if x is odd

Q: ,u(x) -f(x) - I(x) 1.5- 2 3 4 In the figure, is f(x) squeezed by u(x) and 1(x) |? 1. at x = 3? |? 2.…

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Q: Show that if f:R → R is odd and the restriction of fto the interval [0, co) is strictly decreasing,…

A: The odd functions are functions that return their negative inverse when x is replaced with –x. This…

Q: Consider the regular subdivision of the interval [a, b] as a. = x0 < x1 < x2 < x3 < X4 = b, with the…

A: The function f on [a,b] such that f(a)=f(b)=1, fx1=1.5, fx2=fx3=2.The length of the interval [a,b]…

Q: Define f: [0,2]-R by f(x) = x²+2x-1 , then the sup f- inf f is : [0,2] [0,2] 1.a O V8 b 0 3.cO 8 dO

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Q: Suppose f E R[0, 1] and let f() in An k=1 for all n. Prove that an → So f dx.

A: Given -  To Prove -

Q: ral subscript x superscript 0 cube root of u^2-1 d

A: Here we have, f(x)=∫0xu2-13du We have to find f'(3)=?

Q: 3.Show, by means of the example f(t) = sin t, that f ∗ f is not necessarily nonnegative.

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Q: The interval on which f(x) = 3 – 2x – 12x² – x³ is | - concave down is: O (-4, 0) O (-∞, 4) O (4,…

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Q: Assume that f : R → R is defined via the assignment rule f( x ) = e^(2x) . We will use the notation…

A: Step:-1 Process for induction Method:- To show any statement true using induction, we start with…

Q: . Apply Jensen's inequality (2) to f(x) = In x, with i 1 = ..= d = 1/n to prove that 1 Vx1x2...X 0,…

A: Given :-  Apply Jensen's inequality (2) to f  x  = ln x  , with λ1 = λ2 = ..........  = λn = 1n to…

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: Let f(x) = (x" + 10x) e 6x-5, where n is an unspecified constant. Find the value of n such that the…

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Q: Let f(x) = ri- 2xi.Find the value of c that satisfies the conclusion of Rolle's Theorem forf on (0,…

A: The graph is shown below:

Q: If f is differentiable on R and f(0) = 1,|f0(x)| < 1,∀x ∈ R. Show that |f(x)|≤|x|+ 1,∀x ∈ R.

A: Note that |f0(x)| &lt; 1 should have been |f'(x)| ≤ 1. Otherwise, the given inequality won't be…

Q: Find f. f "(x) = x-2, x > 0, f(1) = 0, f(6) = 0 %3D %3D In 6 (In 6) f(x) = -In(u) + u 5

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Q: If f is a function defined from R to R, is given by f(x) = 5x - 3 then f -(x) is given by %3D -- 20…

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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…

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Q: X if x e Q If f (x) is defined on [0,1] by the rule f (x) = < (1-x , if x 4 Q Prove that fof (x) = x…

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Q: The function f(x) =x + 1/x is concave upward on none O (-1) O (0,) O (0,1) O (1,)

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Q: Does the Rolle's Theorem applies for f(z) – ln(2 – z²) over the interval [−1, 1]? O Cannot be…

A: This is a problem related to derivative of the function. Based on the general formula and concept we…

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

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

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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(x)={tan(2x) ax+b if x>0 if x<0 For which values of a and b is f(x) differentiable for every…

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Q: Let f(x) = Q[x] be a irr of prime degree p. If fr non-real roots then th

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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…

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Q: The function f(x) interval [1, e] at x = Hora lo 0 x is the nat

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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: let f [o2]→Rbe afunction if x<1 fex)= Is f Riemaninteglable? 下くx か

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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 f(r) = (In z)e. What is f'(x)?

A: d/dx (f(x)g(x))= f(x)g'(x)+g(x)f'(x)

Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: Suppose f(x) = 4nx?e-Tx -e-n for all x € R. Show that f = -f. %3D

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Q: Let f be the function defined on [0, 1] by when x is irrational, f (x) = 1 when x is rational.…

A: Please see the below picture for detailed solution.

Q: (a) What is F(2) =-| ()dt on (0, 2|? (b) Is F(z) contimuous? (e) Is F'(z) = f(r)? Good Luck

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Q: If f is differentiable and non-negative on an interval I in R then /3f + 1 is differentiable on I.

A: The given statement is true. Because when f will be non negative then the derivative of the given…

Q: 15. If f+g is differentiable at a, are f and g necessarily differentiable at a ? If f g and f are…

A: Let f be any function that is not differentiable  at the point a and define as: gx=-fx for all real…

Q: ) For a, 3 ER and B > 0, let f(x) ={} x° sin(|x|-#), when x # 0 when x = 0 0, ange of a and B (and…

A: Continuity and differentiability

Q: Use : f(z)-L im $(Z+Az)-f(z) AZ to Prave that -1 when w- I Zキ0

A: Here definition of derivative is given in terms of limit. We have to find the derivative of the…

Q: The antiderivative of f(x), denoted by F(x), exhibits an odd symmetry i.e., it satisfies the…

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Q: Evaluate f(322 + z)dz from z = 0 to z = i.

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Q: For the set of functions {sin(x), sin(2x), sin(3x), ...} = {sin(nx)}, n = 1,2,3, ... on the interval…

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Q: if |f] is in L^1(A), then his in L^1(A)?

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Q: Let f be in R[a, b]. Prove that -- f is in R[a, b] and ∫a b -- f(x)dx = -- ∫ab f(x)dx.

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