[tex]e^{x} > 1 + x[/tex] when [tex]e^{c}[/tex] is removed from the expression derived by the mean value theorem.
Let be [tex]f(x) = e^{x}[/tex], for [tex]x > 0[/tex], according to the mean value theorem, also known as Rolle theorem, there is a value [tex]c\in (0, +\infty)[/tex] so that:
[tex]f'(c) = \frac{f(x)-f(0)}{x-0}[/tex] (1)
Where:
[tex]f'(c)[/tex] - First derivative evaluated at [tex]c[/tex].
[tex]f(0)[/tex] - Function evaluated at zero.
Then, we simplify the expression below:
[tex]x\cdot e^{c} = e^{x}-1[/tex]
[tex]e^{x} = 1 + x\cdot e^{c}[/tex]
Please that [tex]1 + x[/tex] is equalized to [tex]e^{x}[/tex] because of [tex]e^{c}[/tex]. If we eliminate [tex]e^{c}[/tex], then we find that [tex]e^{x} > 1 + x[/tex].
We kindly invite to check this question on mean value theorem: https://brainly.com/question/3957181
