How to solve 3^sin^6x + 3^cos^6x by A.m. G.m. Method?

Saturday, May 19, 2012

How to solve 3^sin^6x + 3^cos^6x by A.m. G.m. Method?

Hello!

The AM-GM inequality states that for any non-negative numbers $\displaystyle{a}$ and $\displaystyle{b}$

$\displaystyle\frac{{{a}+{b}}}{{2}}\geq\sqrt{{{a}\cdot{b}}},$ or $\displaystyle{a}+{b}\geq{2}\sqrt{{{a}\cdot{b}}}.$

Therefore it may be suitable for estimating a sum from the below.

Denote $\displaystyle{{\sin}}^{{2}}{\left({x}\right)}={u},$ then $\displaystyle{{\cos}}^{{2}}{\left({x}\right)}={1}-{u}$ and the function becomes

$\displaystyle{{3}}^{{{{u}}^{{3}}}}+{{3}}^{{{{\left({1}-{u}\right)}}^{{3}}}}.$

$\displaystyle{u}$ may be any number in $\displaystyle{\left[{0},{1}\right]}.$

We may apply this for our function and obtain an inequality

$\displaystyle{{3}}^{{{{u}}^{{3}}}}+{{3}}^{{{{\left({1}-{u}\right)}}^{{3}}}}\geq{2}\sqrt{{{{3}}^{{{{u}}^{{3}}}}\cdot{{3}}^{{{{\left({1}-{u}\right)}}^{{3}}}}}}=$

$\displaystyle={2}\cdot{{3}}^{{\frac{{1}}{{2}}{\left({{u}}^{{3}}+{{\left({1}-{u}\right)}}^{{3}}\right)}}}={2}\cdot{{3}}^{{\frac{{1}}{{2}}{\left({1}-{3}{u}+{3}{{u}}^{{2}}\right)}}}.$

The exponent $\displaystyle\frac{{1}}{{2}}{\left({1}-{3}{u}+{3}{{u}}^{{2}}\right)}$ has one and only one minimum at $\displaystyle{u}_{{0}}=\frac{{1}}{{2}}\in{\left[{0.1}\right]},$ the value at $\displaystyle{u}_{{0}}$ is $\displaystyle\frac{{1}}{{8}}.$

So for any $\displaystyle{u}$ we have $\displaystyle{{3}}^{{{{u}}^{{3}}}}+{{3}}^{{{{\left({1}-{u}\right)}}^{{3}}}}\geq{2}\cdot{{3}}^{{\frac{{1}}{{8}}}},$ and this inequality becomes an equality only at $\displaystyle{u}_{{0}}=\frac{{1}}{{2}}.$

Now recall that $\displaystyle{u}={{\sin}}^{{2}}{\left({x}\right)}.$ It is equal to $\displaystyle\frac{{1}}{{2}}$ when $\displaystyle{\sin{{\left({x}\right)}}}=\pm\frac{{1}}{\sqrt{{{2}}}},$ so at $\displaystyle{x}=\frac{\pi}{{4}}+\frac{{{k}\pi}}{{2}}.$ This way we have found minimums of the given function. They are $\displaystyle\frac{\pi}{{4}}+\frac{{{k}\pi}}{{2}},$ and the minimum value is $\displaystyle{2}\cdot{{3}}^{{\frac{{1}}{{8}}}}\approx{2.294}.$

Ninenotes

Saturday, May 19, 2012

How to solve 3^sin^6x + 3^cos^6x by A.m. G.m. Method?

No comments:

Post a Comment

find square roots of -1+2i