We have , $16^{x^{2} + y } + 16^{y^{2}+ x} = 1$ , then we have to find all the real values of $x$ and $y$.I tried this question but i am not able to proceed because I am not able to simplify this expression to an extent that it could be solved.
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1$\begingroup$ It is easy to see that $x$ and $y$ could not be positive, I think. But real numbers, that's hard? $\endgroup$– SawarnikFeb 14, 2014 at 14:13
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7$\begingroup$ One place to start looking is where $x = y$. Then you have $$ 16^{x^2 + x} = \frac{1}{2} \implies x^2 + x = -\frac{1}{4} $$ so $x = y = -\frac{1}{2}$ is one solution (at least now we know that solutions exist). Also, we can easily see that both $x^2 + y$ and $y^2 + x$ must be negative. This limits the search a lot (they're both between $-1$ and $0$). $\endgroup$– ArthurFeb 14, 2014 at 14:15
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$\begingroup$ $16^{x(x-1)}+16^{y(y-1)}=16^{-(x+y)}$ $\endgroup$– LucianFeb 14, 2014 at 14:30
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$\begingroup$ $16^{x^2+y}+16^{x+y^2} = 16^{x^2+y^2}(16^{y-y^2}+16^{x-x^2})$ $\endgroup$– James S. CookFeb 14, 2014 at 14:32
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3 Answers
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$x^2 + y^2 + x + y = (x + 1/2)^2 + (y + 1/2)^2 - 1/2 \geq -1/2$ and equality occurs only when $x = y = - 1/2$.
Using AM-GM inequality $16^{x^2 + y} + 16^{y^2 + x} \geq 2\cdot\sqrt{16^{x^2+y^2+x+y}} \geq 2\cdot16^{-1/4} = 1$ and equality occurs only when $x=y=-1/2$
answered Feb 14, 2014 at 14:43
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$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\down}{\downarrow}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $$ \mbox{Let's}\quad 4^{x^{2} + y} = \cos\pars{\theta}\,,\quad 4^{y^{2} + x} = \sin\pars{\theta}\qquad\mbox{with}\qquad \theta\ \in\ \pars{0,{\pi \over 2}} $$
$$ x^{2} + y ={\ln\pars{\cos\pars{\theta}} \over 2\ln\pars{2}}\,,\qquad y^{2} + x ={\ln\pars{\sin\pars{\theta}} \over 2\ln\pars{2}} $$
$$ \bracks{{\ln\pars{\cos\pars{\theta}} \over 2\ln\pars{2}} - x^{2}}^{2} + x ={\ln\pars{\sin\pars{\theta}} \over 2\ln\pars{2}} $$
$$ x^{4} - {\ln\pars{\cos\pars{\theta}} \over \ln\pars{2}}\,x^{2} + x + \bracks{{\ln^{2}\pars{\cos\pars{\theta}} \over 4\ln^{2}\pars{2}} - {\ln\pars{\sin\pars{\theta}} \over 2\ln\pars{2}}} = 0\,,\qquad \theta\ \in\ \pars{0,{\pi \over 2}} $$ which express $x$ ( parametrically ) as a function of $\theta$.
answered Feb 14, 2014 at 16:17
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$\begingroup$ So how do you find all solutions using that monster function? $\endgroup$– SawarnikFeb 20, 2014 at 20:34
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This is not exactly the original problem , but it is related to a procedure suggested by Arthur. It looks like we can always find real number solutions $(x , y)$ for $A \ge 1$
$$ 16^{x^2 + y} + 16^{y^2 + x} = A $$
$x^2 + y = y^2 + x = \dfrac{ln{(\frac{A}{2})}}{ln{16}}$
answered Feb 14, 2014 at 16:57