MHB Range of k for Real x in $\dfrac{e^x+e^{-x}}{2}\le e^{kx^2}$

[anemone]

Find the range of $k$ for which $\dfrac{e^x+e^{-x}}{2}\le e^{kx^2}$ holds for all real $x$.

 

[chisigma]

Find the range of $k$ for which $\dfrac{e^x+e^{-x}}{2}\le e^{kx^2}$ holds for all real $x$.

[sp]Considering the Taylor expansions...

$\displaystyle \cosh x = 1 + \frac{x^{2}}{2} + ...\ (1)$

$\displaystyle e^{k\ x^{2}} = 1 + k\ x^{2} + ...\ (2)$

... the condition $\displaystyle \cosh x \le e^{k\ x^{2}}, \forall x$ is equivalent to say $\displaystyle k \ge \frac{1}{2}$...[/sp]

Kind regards

$\chi$ $\sigma$

 

[anemone]

[sp]Considering the Taylor expansions...

$\displaystyle \cosh x = 1 + \frac{x^{2}}{2} + ...\ (1)$

$\displaystyle e^{k\ x^{2}} = 1 + k\ x^{2} + ...\ (2)$

... the condition $\displaystyle \cosh x \le e^{k\ x^{2}}, \forall x$ is equivalent to say $\displaystyle k \ge \frac{1}{2}$...[/sp]

Kind regards

$\chi$ $\sigma$

Well done, chisigma! And thanks for participating!(Yes)