Inequalities
Task number: 3051
Prove and remember the following inequalities:
Variant 1
For all \(x \in \mathbb R\), \(e^x \geq 1+x\).
Variant 2
For all \(x \in (-1, \infty)\), \(\ln(1+x) \leq x\).
Variant 3
For all \(x \in (-1,\infty)\), \(1+x \ge e^{\frac {x}{1+x}}\). Equivalently: \(\ln (1+x) \ge \frac {x}{1+x}\).
Variant 4
for all \(x \ge 0\), \(\sin x \le x\).
