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 \(x \ge 0\), \(\sin x \le x\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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