Let f:(a, b) → R be a function and n∈N. Assume that f is n-times differentiable...

Let f:(a, b) → R be a function and n∈N. Assume that f is n-times differentiable and f^(n)(x) = 0 for all x∈(a,b). Show that f is a polynomial of degree at most n−1.

Expert Solution

Colby Messinger answered 2 years ago

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f...

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f is non-decreasing on [a,b] if and only if f′(x) ≥ 0 for all x ∈ (a,b), while if f is non-increasing on [a,b] if and only if f′(x) ≤ 0 for all x ∈ (a, b). can you please prove this theorem? thank you!

Let f : Rn → R be a differentiable function. Suppose that a point x∗ is...

Let f : Rn → R be a differentiable function. Suppose that a point x∗ is a local minimum of f along every line passes through x∗; that is, the function g(α) = f(x^∗ + αd) is minimized at α = 0 for all d ∈ R^n. (i) Show that ∇f(x∗) = 0. (ii) Show by example that x^∗ neen not be a local minimum of f. Hint: Consider the function of two variables f(y, z) = (z − py^2)(z...

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and...

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and such that f(a)=f(b). Then there exists at least one point c on (a,b) such that f'(c)=0." Rolle's Theorem requires three conditions be satisified. (a) What are these three conditions? (b) Find three functions that satisfy exactly two of these three conditions, but for which the conclusion of Rolle's theorem does not follow, i.e., there is no point c in (a,b) such that f'(c)=0. Each...

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f...

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f rises strictly monotonously ⇒ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?) If f is reversable, f has no critical point. (TRUE or FALSE?) If a is a “minimizer” of f, then f ′(a)...

a) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b)...

a) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b) and f ' < -1 on (a, b). Prove that f is strictly decreasing on [a, b]. b) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b) and f ' ≠ -1 on (a, b). Why must it be true that either f ' > -1 on all of (a, b) or f ' < -1 on all...

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show...

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show that there exists a point c ∈ (0, 2π) such that cos(c)f(c) + sin(c)f'(c) = 2 sin(c).

Let A ⊆ R, let f : A → R be a function, and let c...

Let A ⊆ R, let f : A → R be a function, and let c be a limit point of A. Suppose that a student copied down the following definition of the limit of f at c: “we say that limx→c f(x) = L provided that, for all ε > 0, there exists a δ ≥ 0 such that if 0 < |x − c| < δ and x ∈ A, then |f(x) − L| < ε”. What was...

Assume that f is differentiable at a.

Assume that f is differentiable at a.Compute limn→∞n∑i=1k[f(a+in)−f(a)]Deduce the limit limn→∞n∑i=1k[(n+i)mnm−1−1]

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Let f : [a, b] → R be a monotone function. Show that the discontinuity set...

Let f : [a, b] → R be a monotone function. Show that the discontinuity set Disc(f) is countable.

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