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CSS Pure Mathematics QUESTION #1265
Question 1
Using the Mean Value Theorem, which inequality holds for all \(x\in\mathbb{R}\)?
  • \(e^x \leq 1+x\)
  • \(e^x \geq 1+x\)✔️
  • \(e^x = 1+x\) for all \(x\)
  • \(e^x \geq x\) only for \(x>0\)
Correct Answer Explanation
Define \(f(t)=e^t\). By MVT on \([0,x]\): \(e^x - e^0 = e^c \cdot x\) for some \(c\in(0,x)\). Since \(e^c \geq e^0=1\) for \(c\geq0\), we get \(e^x-1\geq x\), i.e. \(e^x\geq 1+x\). A similar argument applies for \(x<0\).