# Genetic Algorithm

Find the value of $\displaystyle x$ that maximizes $\displaystyle \sin^{4}(x)$, $\displaystyle 0 \leq x \leq \pi$ to an accuracy of at least one part in a million. Use a population size of fifty and a mutation rate of $\displaystyle 1/(\text{twice the length of string})$.
So randomly select a population of 50 binary string of length 8. Decode them into base 10. Look at their fitness levels (e.g. $\displaystyle \sin^{4}(x)$). Now exclude $\displaystyle 25$ of the strings with the lowest fitness levels. Use crossover between random pairs of strings to get 25 "child strings." Now use a mutation rate of $\displaystyle 1/16$ on this new population of strings? Because you dont want a population of strings with end digit 0. This will cause domination.
The maximum value of $\displaystyle \sin x$ is $\displaystyle 1$. So the maximum value of $\displaystyle \sin^{4} x$ is $\displaystyle 1$. So maybe use a string length of $\displaystyle 3$? Because $\displaystyle 111 = 7$ in base 10.