3. Let x0, x₁,...,xn be distinct points on the interval [a, b], where Xo

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

(3) Let f: R→ R be defined by f(x) = 5x³ - 7x - 10. Prove that if |x| ≤ 3, then |f(x)| ≤ 166.

Let f : R → R be an increasing function. Show thatf(x) = x, ∀x ∈ R ⇔ f(x) = x, ∀x ∈ Q.

Suppose the function f has the property that IS(x) – SO) < |x - | for each pair of points x, t in the interval (a, b). Prove that f is continuous on (a, b).

3. Give an example of a function f:N→N which is surjective but not injective.

Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = (x – 5)(x – 6)(x – 9), [5, 9] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ± f(b). If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) C = 8.1547,5.8453

Let f be a function defined on [0,1]. Which of the choices is not enough to prove "f(x) >0 for all x on [0,1] is false"? (A) For all x, f(x) <0; (B) There is an x in [0,1] satisfying f(x) <0; (C) There is one x in [0,1] satisfying f(x)=0; (D) f(x) is nonnegative for all x. ABCD

Let n be a positive integer and let f : [0..n] → [0..n] be an injective function. Define the function g : [0..n] → Z as g(x) = n - (f(x))². Prove that is also injective.

Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval. Prove that f(x) and xf(x) are linearly independent on II.