B) Let f:R - R be defined by. 3 if x rational number. f(x) = if x irrational number !3! Show that f is discontinuous at all x in R.

Q: 1. Use the Intermediate Value Theorem to prove that the equation 3x - 4x² = 3 is solvable on the…

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

Q: Suppose h is continuous on ℝ and h(q) = 0 for every rational number q. Does it follow that h(x) = 0…

A:

Q: Show that f(2) =

A: We will show that derivative does not exist using definition. Then we will use geometry.  Calculate…

Q: Show that there is a solution to the following equation in the interval (2,3). (Hint: Use the…

A: Since there are two questions posted, we will answer the first question (1). If you want the second…

Q: Use the Intermediate Value Theorem to show that thefunction f(x) = - 2x2 - 3x + 8 has at least one…

A: fx=-2x2-3x+8; 0,4 Put x=0 into the given function. f0=-202-30+8=8 Put x=4 into the given function.…

Q: What are the zeros of f(x)=2cos(3x)+1 on the interval [0,π]?

A: We have the given function as fx=2cos3x+1. In order to find the zeros of the function fx, we must…

Q: Prove that the function f: N --> Z defined by f(n) = [1 + (-1)^n * (2n - 1)] / 4, is bijective.

A: f: N --> Z defined by f(n) =[1 + (-1)^n * (2n - 1)]/4 We have to prove f is bijective. To show…

Q: 3. Consider a function f(2) that is analytic in a domain that includes the real axis. When z is…

A: Given that,                 fz is an analytic function in a domain that includes the real axis .  We…

Q: Suppose h is continuous on ℝ and h(q) = 0 for every rational number q. Does it follow that h(x) = 0…

A:

Q: 15.C. Show that a rational function (that is, the quotient of two polynomials) is continuous at…

A: I have proved that if g(x) is continuous at a then 1/g(x) is continuous at x=a provides g(a) not…

Q: If f(x) = prove that f (x) is an even fun %3D et-1 2

A: Introduction :

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A:

Q: Prove or disprove that there exist rational numbers x, y EQ such that √11: = x+y√5.

A: Prove or disprove that there exist rational numbers x, y  in Q such that √11 = x+y√5.

Q: 6. Suppose that f(x) = x" + an-1r*-1+ ...+ ao E Z[x]. Ifris rational and x –r divides f(x), prove…

A: We will use division algorithm to prove the fact.

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: Let f(x) be defined for all rational number a in the interval [0, 1]. Let F(x) and F;(x) be defined…

A:

Q: In(2) ith(7-32) + vr² + 1. Answer using interval 3. Find the domain of the function f(x) = + In…

A: In our function three expression added together. Hence we going to see each expression separately…

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

A:

Q: Let g be a function with domain the rational numbers, defined byg(x) = 2/x - √,2for rational x. Is g…

A:

Q: prove using the IVT. √2 exists. Hint: Consider f (x) = x2.

A: we have to show √2 exists. Now let

Q: Determine whether Rolle's Theorem can be applied to 22 –1 f (x) on the closed interval –1, 1. If…

A:

Q: 9. Use the given graph of f (x) x2 to find a number 8 such that %3D if |x – 1| < 8 then |x2 – 1| <;…

A:

Q: Verify that the function, f(x) = x³ – 2x² – 4x + 2, satisfies the three hypotheses of Rolle's…

A:

Q: 1. Let f:R R be defined by {1 if x rational number, 12 if x irrational number f(x) = Show that f is…

A:

Q: if ニ if 1-X fCx +2) = f(x) Hx f on [-3,3] and we Suppose draw the funch'on

A:

Q: and the domain of the function using interval notation. To enter ∞o, type infinity. To enter U, type…

A: We can solve this as follows:

Q: how that: (f)=X*(-f) x(t) is even, then b₁=0, an= To

A:

Q: Verify that the function satisfies the three hypotheses of Rolle as a comma-separated fist.) x) = 3x…

A:

Q: Apply the Intermediate Zero Theorem (IZT) to show that the function f has zeros in between the…

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

Q: Let's define the function f: R+R by f(x) = 4x - 1 2x + 5 (R+ is the positive reals.) Prove that f is…

A:

Q: Q 4. (a). Show that limit *1, does not exist if (5-4x S(x)=. -)- {ax-2 for x <1] for xz1]

A:

Q: Prove thot if yef(x) is continyous and f(x) =0 when x is a ratsonal number, then f(x) =0 for all x €…

A: Given that y = f(x) is continuous and f(x) = 0, x is a rational number. To prove f(x) = 0 for all…

Q: 1, when x is a rational number. f(x)= 0, when x is not a rational number. Determine if the function…

A: Given:  fx=1, when x is a rational0,  when x is not a rational To determine the function  f is…

Q: 4. Verify Rolle's Theorem for f(x) = x² - 2x +1 on [-1,3]

A: Given the function f(x)=x2−2x+1 On the interval [−1,3]

Q: 1 Express the domain of f(x) using interval notation. Use oo for oo and use U to combine 2 - 2 – 3x…

A: We have to find the domain of the given function.

Q: Find the are under the graph of f(x) = 4-x2  on the interval (-2,1)

A: Given, f(x) =4-x2  we have to find the area under the curve on the interval (-2,1)

Q: Use the Intermediate Value Theorem to show that 2х - 3 if – 3 0 has a zero on the interval [-2, In…

A: By using the Intermediate Value Theorem to show that fx=2x-3x+3  if -3&lt;x≤0-3+2 cosh x  if…

Q: Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval.…

A:

Q: 6. Prove from first principles (with ac- 6 argument ) that लेग ½ lim 7--1

A: 1x2+1-12=2-x2+12x2+1                               =2-x2+12x2+12+x2+1…

Q: 4) Consider the function fex)= Slx-s] Evaluate 7-4 for any There is no solution to the equati on…

A: Given: fx=5x-5 so, f7-f47-4=57-5-54-57-4=52-13=513=53

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A:

Q: Find the domain of the function f(x) = √7x − 3 - In interval form, the domain of fis Cubr

A:

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A:

Q: Verify that the function f(x) = x² – 8x + 5 satisfies the three hypotheses of Rolle's Theorem on the…

A:

Q: 13. Use the Intermediate Value Theorem to show that f(x) = 2x° – 3 has a zero in the interval [1,2].…

A: Given:-f(x)=2x3-3 using the intermediate value theoremf(x) is continuous in the interval 1,…

Q: 3-Prove that f(z)=z² analytic

A: Given that f(z)=z2 is analytic.

Q: Do you agree with the contention that the funtions f(x) = x + 2 and g(x) = x2 -  4 ÷ x - 2 are the…

A:

Q: If f(x)=x^3/3 -3x Show that f(x) satisfies the hypothesis of the Roll's Theorem on the interval [-3,…

A: Topic:- rolls theorem

Q: 7. Use the Intermediate Value Theorem (IVT) to show that there is a root of the equation In r = 3 –…

A: We can use intermidiate theorem in order to solve the given problem.

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: