Can u plzz solve this question... Using Lagrange mean value theorem show that e Rest to x > 1 + x, x > 0.

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- Dec 15th 2008, 08:41 PMShivanandLagrange's Mean Value Theorem
Can u plzz solve this question... Using Lagrange mean value theorem show that e Rest to x > 1 + x, x > 0.

- Dec 15th 2008, 11:05 PMbadgerigar
I think you wrote

Quote:

$\displaystyle e^x>1+x, x>0$

- Dec 19th 2008, 04:09 AMvarunnayuduPlease is there any other way
Please sir is there any other weay because this is very confussing .If possible can u give me a reference from where i can get a complete explaination about hw u came to this conclusion....(Bow)

- Dec 20th 2008, 06:31 AMbkarpuz
Clearly, $\displaystyle \mathrm{e}^{x}$ is positive, increasing and convex for $\displaystyle x\geq0$.

Then, for $\displaystyle x\geq0$, the function $\displaystyle \mathrm{e}^{x}$ is always above its tangent lines; i.e., $\displaystyle \mathrm{e}^{x}\geq \mathrm{e}^{x_{0}}(x-x_{0})+\mathrm{e}^{x_{0}}$ for any $\displaystyle x,x_{0}\geq0$ (recall that a tangent line of a function $\displaystyle f$ at the point $\displaystyle x_{0}$ is given by $\displaystyle g(x):=f^{\prime}(x_{0})(x-x_{0})+f(x_{0})$).

Letting $\displaystyle x_{0}=0$, we get the desired result.