6. [-l10 Points] 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 at non-differentiabiiity (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 obiective 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.) Two parts of this question are multiple choice: you only get one submission attempt for those two parts. A peach orchard owner wants to maximize the amount of peaches produced by her orchard. She has found that the pertree yield is equal to 850 whenever she plants 50 or fewer trees per acre, and that when more than 50 trees are planted per acre, the pertree yield decreases by 20 peaches per tree for every extra tree planted. For example, if there were 45 trees planted per acre, each tree would produce 850 peaches. If there were 55 trees planted per acre, each tree would produce 850 - 20 * (55 - 50) peaches, which is roughly equal to 750 peaches. a Find the function that describes the per-tree yield. Y, in terms of x. n:::::::::::mmMmmmwwm w::::::::::emmeWwwm b. Find the total yield per acre. T, that results from planting x trees per acre. n::::::::::mmemmem T = loG' if x is greater than 50 trees per acre c. Differentiate TWith respect to x dT/ dx = l. ifx is less than 50 trees per acre dT/ dx = :153 ifx is greater than 50 trees per acre Does this derivative ever equal zero? 0 No 0 Yes d. Sketch the graph of Tas x varies and hence nd the value ofx that maximizes the yield and the maximum value of the yield. Optimal value of x : trees per acre Maximum yield : peaches per acre Is Tdifferentiable when x equals 50? 0 Yes 0 No