Infi-1 23

Infi-1 23

Intermediate value theorem #theorem

Let f be a continuous function on [a,b]Then f(a)⋅f(b)<0⟹∃x∈[a,b]:f(x)=0

Weierstrass boundedness theorem #theorem

Let f be a continuous function on [a,b]Then ∃m,M:∀x∈[a,b]:m≤f(x)≤MProof:Let f be unbounded from above on [a,b]∀n∈N:∃xn:f(xn)>n[a,b] is bounded⟹By Bolzano-Weierstrass theorem ∃xnk→x:x∈[a,b]f is continuous on [a,b]⟹f(xnk)→f(x)∈Rf(xn)→∞⟹f(xnk)→∞−Contradiction!Similar proof for bounded from below⟹f is bounded on [a,b]

Weierstrass extreme value theorem #theorem

Let f be a continuous function on [a,b]Then ∃c,d∈[a,b]:∀x∈[a,b]:f(c)≤f(x)≤f(d)Proof:f is bounded from above⟹∃M=sup(f(x)) on [a,b]Let n∈NM−1n is not an upper bound of f on [a,b]⟹∃dn∈[a,b]:M−1n<f(dn)∀n∈N:M−1n<f(dn)≤M⟹f(dn)→MBy Bolzano-Weierstrass theorem: ∃dnk→d∈[a,b]f is continuous⟹f(dnk)→f(d)⟹f(dn)→f(d)⟹M=f(d)Similar proof for minimum

Fermat's theorem (stationary points) #theorem

Let f be defined on (a,b)Let x0∈(a,b)If f has a local extremum at x0 and f is differentiable at x0Then f′(x0)=0Proof:Let x be a local minimumf′(x0)=limx→x0f(x)−f(x0)x−x0limx→x0+f(x)−f(x0)x−x0≥0limx→x0−f(x)−f(x0)x−x0≤0limx→x0+f(x)−f(x0)x−x0=limx→x0−f(x)−f(x0)x−x0⟹f′(x0)=0Similar proof for local maximum

Rolle's theorem #theorem

Let f be continuous on [a,b] and differentiable on (a,b)Let f(a)=f(b)Then ∃c∈(a,b):f′(c)=0Proof:f is continuous at [a,b]⟹∃c,d∈[a,b]:∀x∈[a,b]:f(c)≤f(x)≤f(d)…