# What are the critical points of f(x) = sqrt(sqrt(x)e^(sqrtx)-sqrtx)-sqrtx?

Aug 23, 2017

There are none.

#### Explanation:

$f \left(x\right) = {\left({x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}} - {x}^{\frac{1}{2}}$

$f ' \left(x\right) = \frac{1}{2} {\left({x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - {x}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \frac{d}{\mathrm{dx}} \left({x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - {x}^{\frac{1}{2}}\right) - \frac{1}{2} {x}^{- \frac{1}{2}}$

$\textcolor{w h i t e}{f ' \left(x\right)} = \frac{1}{2} {\left({x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - {x}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \left(\frac{1}{2} {x}^{- \frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} + {x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} \left(\frac{1}{2} {x}^{- \frac{1}{2}}\right) - \frac{1}{2} {x}^{- \frac{1}{2}}\right) - \frac{1}{2} {x}^{- \frac{1}{2}}$

Factoring $\frac{1}{2} {x}^{- \frac{1}{2}}$ from the first term:

$f ' \left(x\right) = \frac{1}{4} {x}^{- \frac{1}{2}} {\left({x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - {x}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \left({e}^{{x}^{\frac{1}{2}}} + {x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - 1\right) - \frac{1}{2} {x}^{- \frac{1}{2}}$

Note that ${\left({x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - {x}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} = {\left({x}^{\frac{1}{2}} \left({e}^{{x}^{\frac{1}{2}}} - 1\right)\right)}^{- \frac{1}{2}} = {x}^{- \frac{1}{4}} {\left({e}^{{x}^{\frac{1}{2}}} - 1\right)}^{- \frac{1}{2}}$.

$f ' \left(x\right) = \frac{1}{4} {x}^{- \frac{3}{4}} {\left({e}^{{x}^{\frac{1}{2}}} - 1\right)}^{- \frac{1}{2}} \left({e}^{{x}^{\frac{1}{2}}} + {x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - 1\right) - \frac{1}{2} {x}^{- \frac{1}{2}}$

$\textcolor{w h i t e}{f ' \left(x\right)} = \frac{{e}^{{x}^{\frac{1}{2}}} + {x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - 1}{4 {x}^{\frac{3}{4}} \left({e}^{{x}^{\frac{1}{2}}} - 1\right)} - \frac{1}{2 {x}^{\frac{1}{2}}}$

$\textcolor{w h i t e}{f ' \left(x\right)} = \frac{{e}^{{x}^{\frac{1}{2}}} + {x}^{\frac{1}{2}} {e}^{{x}^{\frac{1}{2}}} - 1}{4 {x}^{\frac{3}{4}} \left({e}^{{x}^{\frac{1}{2}}} - 1\right)} - \frac{2 {x}^{\frac{1}{4}} \left({e}^{{x}^{\frac{1}{2}}} - 1\right)}{4 {x}^{\frac{3}{4}} \left({e}^{{x}^{\frac{1}{2}}} - 1\right)}$

$\textcolor{w h i t e}{f ' \left(x\right)} = \frac{{e}^{\sqrt{x}} + \sqrt{x} {e}^{\sqrt{x}} - 1 - 2 \sqrt[4]{x} {e}^{\sqrt{x}} - 2 \sqrt[4]{x}}{4 \sqrt[4]{{x}^{3}} \left({e}^{\sqrt{x}} - 1\right)}$

Graph this to find its $0$s.

graph{(e^(x^(1/2))+x^(1/2)e^(x^(1/2))-1)/(4x^(3/4)(e^(x^(1/2))-1))-(2x^(1/4)(e^(x^(1/2))-1))/(4x^(3/4)(e^(x^(1/2))-1)) [-10, 10, -5, 5]}

There are no $0$s and the domain of $f '$ is the same as that of $f$, so the function has no critical values.