Let [f:R R] and [g:R R] be two functions defined by [f(x) = _e(x^2 + 1) - e^ - x + 1] and …

Question

Let $$f:R \to R$$ and $$g:R \to R$$ be two functions defined by $$f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$$ and $$g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$$. Then, for which of the following range of $$\alpha$$, the inequality $$f\left( {g\left( {{{{{(\alpha  - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$$ holds ?

Accepted Answer

$$f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$$

$$f'(x) = {{2x} \over {{x^2} + 1}} + {e^{ - x}}$$

$$ = {2 \over {x + {1 \over x}}} + {e^{ - x}} > 0\,\,\forall x \in R$$

$g(x) = {e^{ - x}} - 2{e^x}$

$$g'(x) -  - {e^{ - x}} - 2{e^x}  f\left( {g\left( {\alpha  - {5 \over 3}} \right)} \right)$$

Solutions · 1