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CSS Pure Mathematics QUESTION #4122
Question 1
Using MVT to prove $e^x \geq 1 + x$ for all $x \in \mathbb{R}$, what is the key observation for $x > 0$?
  • $e^c > 1$ for $c \in (0, x)$ implies $\frac{e^x - 1}{x} > 1$✔️
  • $e^c = 1$ for some $c$
  • $f'(x) = e^x$ is decreasing
  • $e^x$ has minimum at $x = 0$
Correct Answer Explanation
For $x > 0$, $c \in (0, x)$ implies $e^c > e^0 = 1$. By MVT: $\frac{e^x - 1}{x} = e^c > 1$, so $e^x - 1 > x$.