Help with Norms

I need help with two exercises:

1. I need to find (with poof) the constants c_1, c_2 > 0, so that the 2 statements / formulas in the first part of the image are true for $\forall x\in\mathbb{R}^n$ :

2. I need to find a constant c > 0, so that for all $x\in \mathbb{R}^n$ the second part of th image (2.) is true and for some $x\neq 0$ , the two  are equal.


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