grandes-ecoles 2020 Q10

grandes-ecoles · France · centrale-maths2__psi Curve Sketching Number of Solutions / Roots via Curve Analysis

Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the inequality with unknown $x \in \mathbb { R }$
$$x \mathrm { e } ^ { x } \leqslant m \tag{I.2}$$
Using the functions $V$ and $W$, determine, according to the values of $m$, the solutions of (I.2). Illustrate graphically the different cases.

Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the inequality with unknown $x \in \mathbb { R }$

$$x \mathrm { e } ^ { x } \leqslant m \tag{I.2}$$

Using the functions $V$ and $W$, determine, according to the values of $m$, the solutions of (I.2). Illustrate graphically the different cases.

Paper Questions

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