(3) (2018A7(b)) Prove that the function f(x) = x² is continuous on any interval [a, b].

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

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

A:

Q: Let T : R → R³ be defined by: X1 T [3x1 – 12 – 5x12x17+ x2] X2 (a) Show that T is a linear…

A:

Q: Determine L[f]if 0, f(1) = {1- 1, 0 2. 1,

A: Given that, A function f(t), t ≥ 0, its Laplace transform F(s) = L{f(t)} is defined as

Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

A:

Q: 13. Let the function f: [a, b] in R be continuous and one-to-one and such that f(a) < f(b). Let c…

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: Let f: Z → Z be defined by f(x) = x² + 1, and let C = {1,2,3}.

A:

Q: Suppose f: R → R is a function defined by f(x) = x³. (a) Find f([-1,1]) and f-¹([0, 8]). (b) Show…

A: We shall use the intermediate value theorem to answer this question

Q: Let f: X --> Y be a function, and let U be a subset of X.  (1) Prove or disprove that f maps X \ U…

A:

Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

A:

Q: O Find the largest value of A such that the function f(x) = x + 1x? - 4x + 6 is decreasing for all x…

A:

Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

Q: b) Prove that if f(z) and f(Z) are analytic in a domain I, then I is a constant function.

A: Given Data: We have to prove  if fz and fz¯ are analytic in a domain Ω, then f is a constant…

Q: Show that if f"(x) < 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] < favg < f[(a+b)/2]

A: Consider the given information. f"(x) &lt; 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] &lt; favg &lt;…

Q: Exercise 2. Let f : A → B be a function with Y1 and Y2 subsets of B. Prove that if Y1 C Y2, then…

A: Let f:A→B be a function and let Y1,Y2⊆B. Suppose that Y1⊆Y2 To show f-1(Y1)⊆f-1(Y2) Let x∈f-1(Y1)…

Q: The function f(x) = x² + 1 satisfies Rolle's Theorem in the interval [-2, –0.5]. - x² Select one: O…

A: