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) = 0. (TRUE or FALSE?) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0. If f is reversable, then f −1 differentiable. (TRUE or FALSE?) If f ′ is limited, then f is lipschitz. (TRUE or FALSE?)

Expert Solution

Here, mainly we use Mean value theorem to justify the statements.

Colby Messinger answered 2 years ago

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 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...

Suppose f : [a, b] −→ R is continuous on [a, b] and that f attains...

Suppose f : [a, b] −→ R is continuous on [a, b] and that f attains its minimum value at ξ ∈ (a, b). If f ' (ξ) exists, prove that f ' (ξ) = 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.

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.

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

Let f(x) be differentiable on the interval [a,b]. Select all of the following statements which must...

Let f(x) be differentiable on the interval [a,b]. Select all of the following statements which must be true. 1. f(x) must attain its maximum value on the interval [a,b] 2. There is some number c in the interval (a,b) such that f′(c)= (f(b)−f(a))/ (b−a) 3. f(x) is integrable on [a,b]. 4. f(x) is continuous on (a,b). 5. f(x) is a polynomial. 6. f(x) is an increasing function.

Problem 2: Let f and g be two differentiable functions defined on an interval (a,b). Assume...

Problem 2: Let f and g be two differentiable functions defined on an interval (a,b). Assume that g(x) dne 0 for all x ∈ (a, b). Prove that f/g is differentiable and (f/g)'(x) = (f'(x)g(x)-f(x)g'(x))/(g^2(x)) for all x ∈ (a, b)

Question

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

Solutions

Expert Solution

Related Solutions

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

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...

Suppose f : [a, b] −→ R is continuous on [a, b] and that f attains...

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

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

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.

Let f(x) be differentiable on the interval [a,b]. Select all of the following statements which must...

Problem 2: Let f and g be two differentiable functions defined on an interval (a,b). Assume...