# What is maximum value of $f\left(x\right)=x\sqrt{x}-\frac{1}{\sqrt{1-x^2}}$?

Find maximum value of $$f\left(x\right)=x\sqrt{x}-\frac{1}{\sqrt{1-x^2}}$$

Domain of the function is $$[0,1)$$. Equating the derivative by zero we get,

$$\frac32\sqrt x-\frac{x}{(1-x^2)^{\frac32}}=0$$ $$\frac94x=\frac{x^2}{(1-x^2)^3}$$ $$9(x^2-1)^3+4x=0$$

Not sure how to find the roots of the equation. But I can proceed as follow,

$$\frac9{4x}=\frac{1}{(1-x^2)^3}$$ $$\frac1{\sqrt{1-x^2}}=\sqrt[\large6]{\frac9{4x}}$$So we are looking for maximum of $$x\sqrt x-\sqrt[\large6]{\frac{9}{4x}}$$

• Actually, $\displaystyle f'(x)=\frac{3 \sqrt{x}}{2}-\frac{x}{\left(1-x^2\right)^{3/2}}$. Sep 12, 2021 at 19:07
• @JoséCarlosSantos Oops! You are right I made a mistake. Sep 12, 2021 at 19:09
• The second part is wrong: the equality is not an identity, so you cannot replace one thing by the other. It is only true at a single point (or maybe a handful of points). Your best best is to solve the equation at the end of the first part numerically. Sep 12, 2021 at 19:41
• I get two solutions, only one of which is in the domain of interest: $x \approx 0.6$. Sep 12, 2021 at 19:45
• Meanwhile, the Galois group of $9(x^2-1)^3+4x$ is $S_6$ Sep 12, 2021 at 20:01

You end with the problem of solving for $$x$$ the sextic equation $$9 x^6-27 x^4+27 x^2+4 x-9=0$$ and, by inspection (as @NickD already commented) the solution is close to $$0.6$$.
Make $$x=y+\frac 35$$ to make $$9 y^6+\frac{162 y^5}{5}+\frac{108 y^4}{5}-\frac{648 y^3}{25}-\frac{1728 y^2}{125}+\frac{53972 y}{3125}+\frac{636}{15625}=0$$ Neglect the terms $$(y^3,\cdots,y^6)$$ and solve the quadratic $$-\frac{1728 y^2}{125}+\frac{53972 y}{3125}+\frac{636}{15625}=0$$ The smallest root is $$y=-\frac{318}{5 \left(13493+\sqrt{183434809}\right)}\implies x=0.597647652\cdots$$ while the exact solution is $$x=0.597647629\cdots$$
Using this approximation, the maximum of the function is then $$-0.78523029529810916\cdots$$ while the exact solution is $$-0.78523029529810836\cdots$$ which is not too bad since obtained at the price of a quadratic equation.