Applications of derivatives: inequalities
Task number: 3022
Prove the following inequalities:
Variant 1
\(e^x \geq x + 1\) for all \(x \in \mathbb R\).
Variant 2
\(\ln(x) \leq x-1\) for \(x \in (0, \infty)\).
Variant 3
\(x + 1 \geq e^{\frac{x}{1+x}}\) for \(x \in (-1, \infty)\).
Variant 4
\(\sin x \leq x\) for \(x \geq 0\).
Variant 5
\(\cos x \geq 1 - \frac{x^2}2\).
