grandes-ecoles 2012 QVI.A

grandes-ecoles · France · centrale-maths1__psi Proof Deduction or Consequence from Prior Results

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Let $g$ be a continuous application from $[0,1]$ to $\mathbb{R}$. We assume that for all $n \in \mathbb{N}$, we have $$\int_0^1 t^n g(t)\,dt = 0.$$
VI.A.1) What can we say about $\displaystyle\int_0^1 P(t)g(t)\,dt$ for $P \in \mathbb{R}[X]$?
VI.A.2) Deduce from this that $g$ is the zero application.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.

Let $g$ be a continuous application from $[0,1]$ to $\mathbb{R}$. We assume that for all $n \in \mathbb{N}$, we have
$$\int_0^1 t^n g(t)\,dt = 0.$$

VI.A.1) What can we say about $\displaystyle\int_0^1 P(t)g(t)\,dt$ for $P \in \mathbb{R}[X]$?

VI.A.2) Deduce from this that $g$ is the zero application.

Paper Questions

QI.A QI.B QI.C QII.A QII.B QII.C QIII.A QIII.B QIII.C QIII.D QIV.A QIV.B QIV.C QIV.D QV.A QV.B QV.C QV.D QV.E QV.F QV.G QVI.A QVI.B QVI.C QVII.A QVII.B QVIII.A QVIII.B QVIII.C QVIII.D QVIII.E QVIII.F QVIII.G