Analysis I

Question 1

Existence Of Zeros [Intermediate Zero]

Answer 1

Let f:[a,b]→ℝ be continuous. If f(a)*f(b) ≤ 0, then there exists c ∈ (a,b) such that f(c) = 0.

Question 2

Intermediate Value Theorem

Answer 2

If f:[a,b]→ℝ is continuous, then f takes all values between f(a) and f(b).

Question 3

Fermat theorem

Answer 3

If f is differentiable at x₀∈ℝ and a neighborhood of x₀ lies in the domain of f, then “x₀ is a local extremum” implies “f′(x₀)=0” (making x₀ a critical point). However, f′(x₀)=0 need not imply x₀ is an extremum.

Question 4

Weierstrass Theorem

Answer 4

If f:[a,b]→ℝ is continuous, then ∃x₁, x₂ ∈ [a,b] such that f(x₁)=sup{f(x) | x∈[a,b]} and f(x₂)=inf{f(x) | x∈[a,b]}.

Question 5

Rolle Theorem

Answer 5

If f:[a,b]→ℝ is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c ∈ (a,b) with f′(c)=0.

Question 6

Mean Value Theorem (Lagrange)

Answer 6

If f:[a,b]→ℝ is continuous on [a,b] and differentiable on (a,b), then ∃c ∈ (a,b) such that f′(c) = (f(b) – f(a)) / (b – a).

Question 7

Cauchy Theorem

Answer 7

If f,g:[a,b]→ℝ are continuous on [a,b], differentiable on (a,b), and g′(x)≠0 on (a,b), then ∃x ∈ (a,b) such that f′(x)/g′(x) = [f(b)–f(a)] / [g(b)–g(a)].

Question 8

Heine–Cantor Theorem

Answer 8

If f:[a,b]→ℝ is continuous, then it is also uniformly continuous on [a,b]. Formally, ∀ε>0, ∃δ>0 such that |x–y|<δ ⇒ |f(x)–f(y)|<ε for all x,y ∈ [a,b].

Question 9

Integral Mean Value Theorem

Answer 9

Define m(f,a,b) = (1/(b–a)) ∫(a→b) f(x)dx as the mean value of f over [a,b]. If f is integrable on [a,b], then inf f(x) ≤ m(f,a,b) ≤ sup f(x).