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 of (a, b)?

Expert Solution

Colby Messinger answered 1 year ago

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)...

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!

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...

a) Suppose f:R → R is differentiable on R. Prove that if f ' is bounded...

a) Suppose f:R → R is differentiable on R. Prove that if f ' is bounded on R then f is uniformly continuous on R. b) Show that g(x) = (sin(x4))/(1 + x2) is uniformly continuous on R. c) Show that the derivative g'(x) is not bounded on R.

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...

Suppose a function f : R → R is continuous with f(0) = 1. Show that...

Suppose a function f : R → R is continuous with f(0) = 1. Show that if there is a positive number x0 for which f(x0) = 0, then there is a smallest positive number p for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) = 0}.)

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.

Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. Define...

Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. Define A := 1/b−a integral of f from a to b, and B := 1/b-a integral of f2 from a to b . Show that 1/b − a integral from a to b of (f(x) − A)2 dx = B − A 2 . Conclude that A2 ≤ B. (b) Assume the Cauchy–Schwarz Inequality for Integrals of Exercise 6.3 #2, which we state here for...

For the following exercises, consider the graph of the function f and determine where the function is continuous/ discontinuous and differentiable/not differentiable.

Question

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

Solutions

Expert Solution

Related Solutions

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

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

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

a) Suppose f:R → R is differentiable on R. Prove that if f ' is bounded...

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

Suppose a function f : R → R is continuous with f(0) = 1. Show that...

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

Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. Define...

For the following exercises, consider the graph of the function f and determine where the function is continuous/ discontinuous and differentiable/not differentiable.

For the following exercises, consider the graph of the function f and determine where the function is continuous/ discontinuous and differentiable/not differentiable.