ROUC - Caractéristiques des fcts - Définitions en langage math

Croissante

• ∀ x₁, x₂ ∈ dom f ; x₁ < x₂ => f(x₁ ) ≤ f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ > x_{2 1} ) ≥ f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ = x₂ => f(x₁ ) = f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ < x₂ => f(x₁ ) ≥ f(x₂ )

Décroissante sur l'intervalle I

• ∀ x₁, x₂ ∈ I ; x₁ < x₂ => f(x₁) ≥ f(x₂)

• ∀ x₁, x₂ ∈ I ; x₁ < x₂ => f(x₁) ≤ f(x₂)

• ∀ x₁, x₂ ∈ I ; x₁ < x₂ => f(x₁) = f(x₂)

• ∀ x₁, x₂ ∈ I ; x₁ < x₂ => f(x₁) ≠ f(x₂)

Paire

• Si ∀x∈domf: -x∈domf et f(-x)=f(x)

• Si ∀x∈domf: x∈domf et f(-x)=f(x)

• Si ∀x∈domf: -x∉domf et f(-x)=f(x)

• Si ∀x∈domf: -x∈domf et f(-x)≠f(x)

Maximum

• ∃a₁ ,a₂∈ℝ, a₁ <a <a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x) ≤ f(a)

• ∃a₁ ,a₂∈ℝ, a₁ >a >a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x) ≥ f(a)

• ∃a₁ ,a₂∈ℝ, a₁ <a <a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x) = f(a)

• ∃a₁ ,a₂∈ℝ, a₁ <a <a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x) ≥ f(a)

Minimum

• ∃a₁ ,a₂∈ℝ, a₁ <a <a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x)≥f(a)

• ∃a₁ ,a₂∈ℝ, a₁ >a >a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x)≤f(a)

• ∃a₁ ,a₂∈ℝ, a₁ <a <a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x)=f(a)

• ∃a₁ ,a₂∈ℝ, a₁ <a <a₂ tq ∀x∈]a₁,a₂[⋂dom f, f(x)≠f(a)

Décroissante

• ∀ x₁, x₂ ∈ dom f ; x₁ < x₂ => f(x₁ ) ≥ f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ > x₂ => f(x₁ ) ≤ f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ = x₂ => f(x₁ ) = f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ < x₂ => f(x₁ ) < f(x₂ )

Minimum absolu

• ∀x∈dom f,f(x)≥f(a)

• ∀x∈dom f,f(x)≤f(a)

• ∃x∈dom f,f(x)=f(a)

• ∀x∈dom f,f(x)=f(a)

Strictement décroissante

• ∀ x₁, x₂ ∈ dom f ; x₁ < x₂ => f(x₁ ) > f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ > x₂ => f(x₁ ) < f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ = x₂ => f(x₁ ) = f(x₂ )

• ∀ x₁, x₂ ∈ dom f ; x₁ < x₂ => f(x₁ ) = f(x₂ )

Croissante sur l'intervalle I

• ∀ x₁, x₂ ∈ I ; x₁ < x₂ => f(x₁ ) ≤ f(x₂ )

• ∀ x₁, x₂ ∈ I ; x₁ > x₂ => f(x₁ ) ≥ f(x₂ )

• ∀ x₁, x₂ ∈ I ; x₁ = x₂ => f(x₁ ) = f(x₂ )

• ∀ x₁, x₂ ∈ I ; x₁ < x₂ => f(x₁ ) ≠ f(x₂ )

Maximum absolu

• ∀x∈dom f,f(x)≤f(a)

• ∀x∈dom f,f(x)≥f(a)

• ∀x∈dom f,f(x)=f(a)

• ∀x∈dom f,f(x)≠f(a)