The minimum value of 27^cos x + 81^sin x is?
Hello,
Yes !
$\displaystyle 27=3^3$ and $\displaystyle 81=3^4$
Therefore $\displaystyle 27^{\cos(x)}+81^{\sin(x)}=3^{3 \cos(x)}+3^{4 \sin(x)}$
By AM-GM :
$\displaystyle 3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}$
So you just have to find the minimum of $\displaystyle 3 \cos(x)+4 \sin(x)$
This is periodic with perion $\displaystyle 2 \pi $, so sketch the curve. This has a minimum near $\displaystyle x=4$. The minimum is a root of:
$\displaystyle f(x)=\ln(81) \cos(x) e^{\ln(81) \sin(x)}-\ln(27) \sin(x) e^{\ln(27) \cos(x)}$
Running the bisection method over this gives the minimum occurs at $\displaystyle x \approx 3.86147$, where the function is very close to $\displaystyle 0.141494$ .
RonL
Then we can say that the minimum of the function will always be superior to the minimum of 3cos(x)+4sin(x)
But with the large inequality, I don't know, maybe it can be equal.
No, because it's 3 to the power -5.@Moo: Minimum of 3 cos x + 4 sin x = - 5.
Gotta think about that thenThe answer is 1/4.
It's ok, I figured it out !
There is equality in AM GM if $\displaystyle 3^{3 \cos(x)}=3^{4 \sin(x)}$
So let m be the minimum value of $\displaystyle 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}$, that is $\displaystyle \frac 23 \sqrt{\frac 13}$ (substituting -5).
$\displaystyle 3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m$
If $\displaystyle 3^{3 \cos(x)}=3^{4 \sin(x)}$, then $\displaystyle 3^{3 \cos(x)}+3^{4 \sin(x)}=2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m$
Thus m is the minimum, in particular when $\displaystyle 3^{3 \cos(x)}=3^{4 \sin(x)}$
Sounds more correct ?
Edit : I have to work it out deeper, because I assumed that 3cos(x)=4sin(x)