Question

Alana consulted a local master beekeeper, Dr. Beekeeper, concerning the death of some of her beehives. He suggested adding a specific type of plant food, “Bee Empower” to the clover fields where the bees forage for nectar. The amount of Bee Empower (in kg) that should be added per acre of clover field is represented by:

f(x)= -9x² + 126x -45

x=kg of Bee Empower/acre

A. Find the critical value of Bee Empower for this function

B. In words, describe what this value means

C. Is this value the minimum or maximum amount of plant food needed per acre? Mathematically prove your answer.

Answer 1

Given where x denotes the amount of Bee Empower (in kg) that should be added per acre of the clover field

A) At the critical point, the derivative of a function will be zero

Therefore the critical value of the Bee Empower for this function is 7

B) The critical points are the values where the function f(x) doesn't change with respect to a small change in the value x. Also at critical points, the function will have a maximum or minimum value.

C) f(x=6) = 387, f(x=7) = 396, f(x=8) = 387

Therefore 7 is the maximum amount of pant food needed per acre.

Mathematical proof:

We use the second derivative to find the shape of the curve at the critical point. If it is negative then the curve is concave down at critical point and vice-versa.

==> The curve is concave down. Therefore the critical point 7 is the maximum value.