## Minimizing a function

f <- function(x) x^2 + 1

x <- -10:10

plot(x, f(x))

## pdf('../figs/quad.pdf')
## plot(x, f(x))
## dev.off()

## We can easily see that the minimum is at zero
## At the value x = 0 the function takes the value of 1

## How do we use tools in R to minimize this function?

## Use the function optimize

op <- optimize(f = f, lower = -20, upper = 20)

## Where is the maximum of this function? 

f2 <- function(x) 6 + x * 7.5 + x^2 * -0.6

x <- -20:20

plot(x, f2(x))

## It is not easy to determine visually

op2 <- optimize(f = f2, lower = -20, upper = 20, maximum = TRUE)

## Analytical method
## Derive and equate to zero

df2 <- function(x) 7.5 + x * 2 *-0.6

## Answer 

max.df2 <- -7.5/(2 * -0.6)

# A third function

f3 <- function(x) log(x) / (1 + x)

## This function implies that the domain set is x >= 0
## There is no analytical solution to find the maximum

xx <- seq(1, 5, 0.05)

plot(xx, f3(xx))

df3 <- function(x) (1 + 1/x - log(x))/(x + 1)^2

plot(df3(xx) ~ xx)
abline(h = 0)

op3 <- optimize(f = f3, lower = 1, upper = 5, maximum = TRUE)