31-The annual dry matter production of a certain wheat variety y (g/m²), as a function of the average annual precipitation x, is given by the function: y (x) = 0.000352 – 0.000611x + 0.00032x² - 0.000284x³ Find the value of the average annual precipitation at which dry production increases.

"The dry matter production of the wheat variety as a function of average annual precipitation is a cubic function. To find the value of precipitation at which the dry production increases, we need to determine the critical points of the function, which are the values of x where the first derivative y'(x) is equal to zero. The points where y'(x) changes from negative to positive correspond to local minima, which indicates that the dry production starts to increase.

The first derivative y'(x) is given by:
y'(x) = -0.000611 + 2 \cdot 0.00032x - 3 \cdot 0.000284x^2

Setting y'(x) equal to zero gives us the critical points:
-0.000611 + 0.00064x - 0.000852x^2 = 0

Let's solve this quadratic equation to find the values of x .

The critical points for the function, calculated from the derivative, are complex numbers. Since precipitation must be a real number, this suggests that the cubic function does not have real extrema within a physically meaningful range for precipitation.

However, it's important to note that the behavior of a cubic function means that there will always be a point where the function changes from decreasing to increasing (or vice versa), and this occurs at an inflection point rather than a minimum or maximum. To find the inflection point where the production starts to increase, we need to find where the second derivative changes sign.

The second derivative of the production function is given by:
y''(x) = 2 \cdot 0.00064 - 6 \cdot 0.000284x

Setting y''(x) equal to zero will give us the inflection point. Let's calculate that.

The value of the average annual precipitation at which the dry matter production starts to increase is approximately 0.751 \, \text{m}^2 . This is the inflection point of the function."