grandes-ecoles 2018 Q20

grandes-ecoles · France · x-ens-maths__psi Inequalities Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.)
Show that $\lambda ^ { \alpha } \leq 1 + \lambda$ for all real $\lambda > 0$ and deduce the inequality:
$$\left| x - \frac { k } { n } \right| ^ { \alpha } \leq n ^ { - \alpha / 2 } \left( 1 + \sqrt { n } \left| x - \frac { k } { n } \right| \right)$$
for all $x \in ]0,1[ , n \in \mathbb { N } ^ { * }$ and $k \in \{ 0 , \ldots , n \}$. 
Show that $\lambda ^ { \alpha } \leq 1 + \lambda$ for all real $\lambda > 0$ and deduce the inequality:

$$\left| x - \frac { k } { n } \right| ^ { \alpha } \leq n ^ { - \alpha / 2 } \left( 1 + \sqrt { n } \left| x - \frac { k } { n } \right| \right)$$

for all $x \in ]0,1[ , n \in \mathbb { N } ^ { * }$ and $k \in \{ 0 , \ldots , n \}$.
Paper Questions
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