Question

Une fonction polynomiale du second degré possède les caractéristiques

suivantes :
• La fonction est croissante sur l'intervalle [0, 8].
• La fonction est décroissante sur l'intervalle ]-10, 0].
• La courbe passe par le point (-8, 4).
PSST
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SAVOIR-FAIRE
INDICE
Représente graphiquement la fonction, puis détermine son codomaine. Utilise la
notation décimale au besoin.
Le codomaine est.

Je n’arrive pas à trouver le co domaine si quelqu’un peut m’aider svp?

Answer 1

A second-degree polynomial takes the form

`f(x) = ax^2 + bx + c`

for some constants a, b, and c. Such a function is continuous and differentiable everywhere in its domain.

Differentiating f(x) with respect to x gives

`(df)/(dx) = 2ax + b`

We're told that

• f(x) is increasing on [0, 8]

• f(x) is decreasing on ]-10, 0]

and since df/dx is continuous, this tells us that df/dx = 0 when x = 0. It follows that b = 0, and we can write

`f(x) = ax^2 + c`

We're also given that the curve passes through the point (-8, 4), which gives rise to the constraint

`f(-8) = 64a - 8b + c = 4 \implies 64a + c = 4`

Since f(x) is decreasing on ]-10, 0], we have df/dx < 0 when x = -8, so

`2ax+b < 0 \implies -16a < 0 \implies a > 0`

This means f(x) is minimized when x = 0, with min{f(x)} = c, so the co-domain of f(x) is the set {f(x) ∈ ℝ : f(x) ≥ c}.

Without another condition, that's all we can say about the co-domain. There are infinitely many choices for the constants a and c that satisfy the given conditions. For example,

a = 1, c = -60 ⇒ f(x) = x² - 60 ⇒ f(-8) = 4

⇒ co-domain = {f(x) ∈ ℝ : f(x) ≥ -60}

a = 2, c = -124 ⇒ f(x) = 2x² - 124 ⇒ f(-8) = 4

⇒ co-domain = {f(x) ∈ ℝ : f(x) ≥ -124}

etc.