Let f € C[a, b]. Suppose that for every [a1, b1] C [a, b], S f(x) dx = 0. Show that f(x) = 0, x E [a, b].

Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: Let f(e) = In[r*(z + 6}*(z° + 6)"] f'(x) =

A: Given: Now,         f'(x)=lnx4x+62x2+69'        =1x4x+62x2+69x4x+62x2+69'…

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A: To prove: G(x) = F(x) + C Given that F is an antiderivative of f on an interval I, that is F'=f then…

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Q: Define h(2) in a way that extends h(t) = (t2 + 3t - 10)/(t - 2) to be continuous at t = 2.

A: Consider the given the function,

Q: 2) Find a value of c for which f (x) is a pdf on the f(x) = cx + x², [0. 1]

A: Introduction ∫-∞∞fxdx=1 The total area under the curve fx of pdf is 1.

Q: If f has a finite third derivative f ’‘’ in [a, b] and f(a) = f‘(a) = f(b) = f’(b) = 0, then there…

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Q: Find a and b such that f(x) is continuous at x=1. [x – 2] + a x 1

A: A detailed solution is given below.

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: Let f(x) = x – x³ on [0, 1]. f(0) f(1) = Find c such that f'(c) = 0 or, if Rolle's Theorem does not…

A:

Q: Let $X$ have the pdf $f(x)=4 x^{3}, 0&lt;x&lt;1,$ zero elsewhere. Find the cdf and the pdf of…

A: GivenThe pdf of X is f(x)=4x3                       , 0&lt;x&lt;10                           ,…

Q: Assume that the function f is such that f(3) = 1, ſ f(x) dx = 5, and f'(x) is continuous on [0, 3].…

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

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Q: For what value of b is

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Q: 2. If f is a continuous function such that S(t)dt = cos(2x)e* - [e s(t)dt for all x, find an…

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Q: iii. is continuous at a provided g(a) # 0.

A:

Q: Let g(x) = kf(x) for all x E/= {x: asxsb}. Show that Jglx)dx = kJ f(x)dx if k<0.

A: To prove the above result we use the definition of lower and upper Riemann integrals. For the proof…

Q: Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x ≥ 0…

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Q: Suppose that g and h are continuous functions and that S8(x) dx = 5, h(x)dx =-1, and h(x) dx = 3.…

A: There is no such rule of division of two integral.

Q: If f is continuous on [a,b], and f(x) > 0 then J VFX)dx = = /J f(x)dx | F(x)dx (A) True B) False

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Q: If f(x) = t°dt then = (x),f

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Q: If L {f(t)} = F(s) then L{eat f(t)} = F(s) (s-a) F (s) s-a F (s-a) F (s + a) خيار 2 الخيار رقم 1

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Q: Let f (x) belong to Zp[x]. Prove that if f (b) = 0, then f (bp) = 0.

A:

Q: If x is ar.v. with c.d.f. F(x), F(x) is continuous at x = 3, then p(x = 3) = 0 %3D O True False

A: It is given that the CDF of x at x = 3 is continuous.

Q: ) For what value of a is fx² -1 x<3 S(x)= { |2ax x23 Continuous at every x?

A:

Q: O Let f : A - B be an injection, and X, Y CA. Show that f(Xn Y) = f(X)nf(Y). |

A:

Q: (a) Prove that if m ₹ N, then the function f(x) = xm is continuous on R.

A: We have to prove that, the function f(x) = xm,   m∈ℕ  is continuous on ℝ.

Q: If f(x) is continuous on a, b] thenF(x) = So f(t)dt is continuous. OTrue False

A: We need to check whether the given statement is true or false

Q: Find the Wronskian for the set of functions.{ex, e−x}

A: Given set of functions ,f1=ex &amp; f2=e-x

Q: If Tn,0(x) = ao +a¡x+ a2x² +.. he function f(x) centered at x = %3D -) True p) False

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

Q: If f is continuous on [0, 1], prove that f₁¹ f(x) dx = f₁¹ ƒ(1 − x)dx.

A: This question can be solved using properties of definite integrals.

Q: If f(1) = 2, f(4) = 7, f'(1) = 5, f'(4) = 3, and f"(x) is continuous, what is xf"(x) dx? %3D %3D

A: In the question it is asked to solve for the initial value integration given in question.

Q: f is continuous and f(x) dx = 8, find xfc

A: we have to find

Q: If f and g are continuous on a <x< b, then S) + g(x)] dx = Sf(x)dx - g(x)dx %3D

A: Here I am attaching image so that you understand each and every step.

Q: Let f(x) x² and g(x) = xf(x). Find c = (0, 1) as guaranteed by the Cauchy Mean Value Theorem.

A:

Q: 4. Let f € L'(R). Prove that the function F(x) = S f(t)dt is continuous.

A: Let, f∈L1ℝ. Since,  L1=f:ℝ→ℂ∫ℝftdt&lt;∞∫ℝftdt≤∫ℝftdt&lt;∞Therefore,∫ℝftdt&lt;∞ ∫ℝftdt is a finite…

Q: Let A(z) – S. f(t) dt, with f(x)

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Q: - For what value of b is Sx, bx?, x2 -2 x<-2 х, g(x) = continuous at every x?

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Q: Consider the functions f(x) = and g(r) = a in L(0, 1), with w(x) = x. %D %3D Compute || f | and (f,…

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Q: Find the Wronskian for the set of functions.{e−x, xe−x, (x + 3)e−x}

A: To Find theWronskianfor the set of functions. {e-x,xe-x,(x+3)e-x}

Q: | For what value of a is Sx² – 1, x< 3 |2ax, f(x) = continuous at every x?

A: Let f is continuous at x = 3 then left and right limit are equal.

Q: If ƒ(x) is continuous [-00,b]then on ſf(x)dx= lim [ ƒ (x)dx а. a a

A: Explanation of the solution is given below...

Q: Let A(x) = f(t)dt, with f(x) shown in the graph below.

A: Given that: Graph of f(x)

Q: If X is a continuous RV, prove that E(X) = [1- F(x)] dx – F(x) dx. %3D

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Q: Find the Wronskian of the set of functions {e*,e® sin(x)}.

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Q: For all continuous function f: [0, +o] → R, a > 0 we need to prove that 2*f) dx = [ xfw)dx

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Q: If f is a continuous function satisfying f(x) = x² - x +(1- x)[, fodi+xf (1-r*)f()dt for all x, then…

A: We have to find it using various differential calculus techniques.