A math problem of Liberal Arts in senior two. (Application of derivative) A manufacturer manufactures and sells a kind of beverage in spherical bottles. The manufacturing cost of the bottle is 0.8 π R ^ 2, where R is the radius of the bottle, and the unit is cm. It is known that the manufacturer can make a profit of 0.2 points for every 1ml of beverage sold, and the maximum radius of the bottle that the manufacturer can manufacture is 6cm Q: (1) what is the radius of the bottle to maximize the profit of each bottle? (2) When the bottle radius is large, the profit of each bottle is the smallest? Please write the process, thank you

A math problem of Liberal Arts in senior two. (Application of derivative) A manufacturer manufactures and sells a kind of beverage in spherical bottles. The manufacturing cost of the bottle is 0.8 π R ^ 2, where R is the radius of the bottle, and the unit is cm. It is known that the manufacturer can make a profit of 0.2 points for every 1ml of beverage sold, and the maximum radius of the bottle that the manufacturer can manufacture is 6cm Q: (1) what is the radius of the bottle to maximize the profit of each bottle? (2) When the bottle radius is large, the profit of each bottle is the smallest? Please write the process, thank you

Let the profit be YY = 4 π R ^ 3 / 3 * 0.2-0.8 π R ^ 2, so y '= 0.8 π R ^ 2-1.6 π R 0 < R ≤ 6, let y' = 0 get r = 2, then the function y is a monotone decreasing function in (0,2)] and a monotone increasing function in [2,6]. Then the profit is the smallest when r = 2, y = - 16 π / 15 when r = 0, the profit is y = 0, when r = 6, y = 144 π / 5, so the bottle radius is 6cm