Question

Use the Mean Value Theorem to establish the fact that, for x>0,ex>1+x . (DO NOT use any other method).

Answer 1

$e^{x} > 1 + x$ when $e^{c}$ is removed from the expression derived by the mean value theorem.

Let be $f(x) = e^{x}$, for $x > 0$, according to the mean value theorem, also known as Rolle theorem, there is a value $c\in (0, +\infty)$ so that:

$f'(c) = \frac{f(x)-f(0)}{x-0}$ (1)

Where:

$f'(c)$ - First derivative evaluated at $c$.

$f(0)$ - Function evaluated at zero.

Then, we simplify the expression below:

$x\cdot e^{c} = e^{x}-1$

$e^{x} = 1 + x\cdot e^{c}$

Please that $1 + x$ is equalized to $e^{x}$ because of $e^{c}$. If we eliminate $e^{c}$, then we find that $e^{x} > 1 + x$.

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