x is number of trees per acre and y is yield in bushels per acre

For x<40 y=25x

For x≥40

y=(25-1/2(x-40))x

y=(25-1/2x+20)

y=-x^{2}/2+45x

Maximum where derivative is zero or an end point

dy/dx=-x+45

0=-x+45

x=45

end points are x=40 or x=90

test these or do first or second derivative test

Test values

y=-(40)^{2}/2+45(40)=1000

y=-(45)^{2}/2+45(45)=1012.5

y=-(90)^{2}/2+45(90)=0

Absolute max x=45

First derivative test:

test to the left and right of critical point to see if it is a max or min

less than 45 (but more than 40 because it must be in the domain)

x=43

dy/dx=-43+45=2 derivative is positive when x is less than 45

more than 45 (but less than 90 because it must be in the domain)

x=50

dy/dx=-50+45=-5 derivative is positive when x is more than 45

because the derivative is positive from 40 to 45 yield is increasing from x=40 to x=45

because the derivative is negative from 45 to 90 yield is increasing from x=45 to x=90

This means x=45 is a local max and since there are no other critical values then it is higher than the two endpoints

Second derivative test

d^{2}y/dx^{2}=-1

second derivative is negative so that means y is concave down so x=45 (the critical point above) is a local maximum. since there are no changes in concavity and only one critical value x=45 is the absolute max