Step-by-step explanation:
Find the function that describes the per-tree yield, Y, in terms of x.
A) Y = 1100 ... if x is no more than 45 trees per acre
B) Y = 1100 - 30*(x-45) = 2450 - 30x if x is greater than 45 trees per acre
Find the total yield per acre, T, that results from planting x trees per acre.
C) T =1100*x ...if x is no more than 45 trees per acre
D) T = (2450 - 30x)*x = 2450x - 30x² if x is greater than 45 trees per acre
Differentiate T with respect to x
E) dT / dx =1100 ...if x is less than 45 trees per acre
F) dT / dx = 2450 - 60x ...if x is greater than 45 trees per acre
G) Does this derivative ever equal zero?
The first part 1100 is never equal to zero while
2450 - 60x = 0 ==> x = 2450/60 = 40.833 < 45
so dT/dx is never equal to zero.
Moreover when x>45, 2450 - 60x <0 .
So the function T is increasing for x <45 and decreasing
for x>45. The maximum can be at x = 45.
H) Optimal value of x :45 trees per acre
I) Maximum yield : 1100*45 = 49500 peaches per acre
By the way, the same yield is 2450*45 - 30*45² = 49500.
J) Is T differentiable when x equals 45?
T is not differentiable at x = 45, because
(x--> 45 - ) lim dT/dx = 1100
(x--> 45 + ) lim dT/dx = 2450 - 60*45 = -250
1100=/= -250