Final answer:
The rate of change of photosynthesis with respect to the intensity can be found by differentiating the equation. When x = 1, the rate of change is 200, and when x = 3, the rate of change is 60. The rate of change of the rate found in part (i) is 110 when x = 1 and -280 when x = 3.
Step-by-step explanation:
The photosynthesis equation given is f(x) = 145x² - 30x³.
(i) To find the rate of change of photosynthesis with respect to the intensity, we need to differentiate the equation with respect to x. Taking the derivative of the equation, we get f'(x) = 290x - 90x².
(ii) To find the rate of change when x = 1 and x = 3, we substitute these values into the derivative equation. When x = 1, the rate of change is f'(1) = 290(1) - 90(1)² = 290 - 90 = 200. When x = 3, the rate of change is f'(3) = 290(3) - 90(3)² = 870 - 810 = 60.
(iii) To determine how fast the rate found in part (i) is changing when x = 1 and x = 3, we differentiate the derivative equation with respect to x. Taking the second derivative, we get f''(x) = 290 - 180x. Substituting x = 1 and x = 3 into the second derivative, we find that the rate is changing at a rate of f''(1) = 290 - 180(1) = 110 when x = 1, and f''(3) = 290 - 180(3) = -280 when x = 3.