Question

For what value of $x$ does the expression $2^{x^3}-1+\frac{1}{2^{x^3}+2}$ assume the least value.

Answer 1

let $2^{x^3}=t$

$f\left(t\right)=t-1+\frac{1}{t+2}=(t-1)+(t+2{)}^{-1}$

$f^{\prime }\left(t\right)=1-(t+2{)}^{-2}$

$f^{\prime }\left(t\right)=0$

$1-(t+2{)}^{-2}=0$

$1=(t+2{)}^2$

$t=1ort=-3$

but $2^{x^3}>0$

so $t\ne -3$

$t=1$

$2^{x^3}=1$

$x^3=0$

$x=0$

$leastvalue=1-1+\frac{1}{3}$=$\frac{1}{3}$