This question makes an important point: the maximum point of a function may not always be found by solving f '(x) = 0. Remember: functions can have their minimum or maximum at an endpoint of their domain, at a point of non-differentiability (think of the absolute value function, which has its minimum point at zero) or may not even have a maximum or minimum. This means that the most thorough way of solving optimization problems involves sketching the objective function. (For questions that appear on tests, however, the optimum will usually occur at a relative max. or relative min. that can be found by solving f '(x) = 0.)   A peach orchard owner wants to maximize the amount of peaches produced by his orchard. He has found that the per-tree yield is equal to 950 whenever he plants 55 or fewer trees per acre, and that when more than 55 trees are planted per acre, the per-tree yield decreases by 20 peaches per tree for every extra tree planted. For example, if there were 50 trees planted per acre, each tree would produce 950 peaches. If there were 60 trees planted per acre, each tree would produce 950 - 20 * (60 - 55) peaches, which is roughly equal to 850 peaches.   Find the function that describes the per-tree yield, Y, in terms of x. Y =  _____ if x is no more than 55 trees per acre Y =  _____ if x is greater than 55 trees per acre Find the total yield per acre, T, that results from planting x trees per acre. T = _____ if x is no more than 55 trees per acre T =  _____ if x is greater than 55 trees per acre Differentiate T with respect to x dT / dx = _____ if x is less than 55 trees per acre dT / dx = _____ if x is greater than 55 trees per acre Does this derivative ever equal zero? No Yes     Sketch the graph of T as x varies and hence find the value of x that maxi

This question makes an important point: the maximum point of a function may not always be found by solving f '(x) = 0.

• Remember: functions can have their minimum or maximum at an endpoint of their domain, at a point of non-differentiability (think of the absolute value function, which has its minimum point at zero) or may not even have a maximum or minimum.
• This means that the most thorough way of solving optimization problems involves sketching the objective function. (For questions that appear on tests, however, the optimum will usually occur at a relative max. or relative min. that can be found by solving f '(x) = 0.)

A peach orchard owner wants to maximize the amount of peaches produced by his orchard.

He has found that the per-tree yield is equal to 950 whenever he plants 55 or fewer trees per acre, and that when more than 55 trees are planted per acre, the per-tree yield decreases by 20 peaches per tree for every extra tree planted.

For example, if there were 50 trees planted per acre, each tree would produce 950 peaches. If there were 60 trees planted per acre, each tree would produce 950 - 20 * (60 - 55) peaches, which is roughly equal to 850 peaches.

1. Find the function that describes the per-tree yield, Y, in terms of x.

   Y =  _____ if x is no more than 55 trees per acre

   Y =  _____ if x is greater than 55 trees per acre

2. Find the total yield per acre, T, that results from planting x trees per acre.

   T = _____ if x is no more than 55 trees per acre

   T =  _____ if x is greater than 55 trees per acre

3. Differentiate T with respect to x

   dT / dx = _____ if x is less than 55 trees per acre

   dT / dx = _____ if x is greater than 55 trees per acre

   Does this derivative ever equal zero?

   No
   Yes    
4. Sketch the graph of T as x varies and hence find the value of x that maximizes the yield and the maximum value of the yield.

   Optimal value of x : ____ trees per acre

   Maximum yield :  ____ peaches per acre

   Is T differentiable when x equals 55?

   Yes
   No