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  • Hey I want the easiest method to prove $\exp(x)\geq x+1$.

The only method I use is to consider a new function $F$, that realizes $F(x)=\exp(x)-x-1$ then calculate the derivative then use its monotony to prove that $F(x)<0$ I'm only a high school student, if you could use function study it would be easier for me to understand.
So do you have any better method? this one takes me some time to write down I would love a easier method

Stackman
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Bénz Anas
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$$\exp(x) = 1 + \int_0^{x} \exp(t) \ \mathrm dt > 1 + \int_0^x 1 \ \mathrm dt = 1 + x$$

Kenny Lau
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  • That's what i was looking for! Beautiful method,thanks for taking the time to share it with me i'm grateful. – Bénz Anas Jan 01 '19 at 10:51
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Defining $$f(x)=e^x-x-1$$ then $$f'(x)=e^x-1$$ and $$f''(x)=e^x>0$$ Can you proceed?

Dr. Sonnhard Graubner
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    Sometimes I wonder how much of the questions you read. OP wrote: "The only method i use is to consider a new function F, that realizes F(x)=exp(x)-x-1 then calculate the derivative then use its monotony ... So do you have any better method?" – Martin R Dec 31 '18 at 18:50