Problem $1.$ Dose$-$Response of Multi$-$Vitamins Multi$-$vitamins typically have dose$-$response curves of the following form: $\backslash [$ R$=$f$($x$)=\backslash$frac$\{$a x$\}\{$k$^\{2\}+$x$^\{2\}\},\backslash$quad x $\backslash$geq $0\backslash ]$ where $\backslash ($ x $\backslash )$ is a measure of the daily dose, $\backslash ($ R $\backslash )$ is a measure of the health benefits, and $\backslash ($ a $\backslash )$ and $\backslash ($ k $\backslash )$ are positive constants. a$)$ According to this model what is the health benefit to taking no vitamins? How does this change if you take a very small amount? b$)$ What happens if you take an extremely large dose? c$)$ What is the interval of dosage where taking more will increase the benefit? What is the interval of dosage where taking more will decrease the benefit? d$)$ Is there ever a negative benefit to taking the vitamins $($according to this model$)?$ e$)$ Is there a dosage that maximizes or minimizes the health benefits? f$)$ Choose positive constants $\backslash ($ a $\backslash )$ and $\backslash ($ k $\backslash ),$ and graph the function. Interpret your graph in terms of the health benefits of the multi$-$vitain.