Harvest Function and Critical Numbers

The video tutorial discusses a farmer's problem of maximizing fruit yield by adjusting the number of trees planted per acre. Initially, 50 trees yield 65 bushels each, but adding more trees decreases yield per tree by 3 bushels. The tutorial formulates a quadratic function for total harvest, H(x), and solves it to find the optimal number of trees. By calculating the derivative and using the second derivative test, it confirms that planting 36 trees per acre maximizes the harvest. The video concludes with a verification of the solution and emphasizes the importance of understanding quadratic functions in optimization problems.

What is the initial yield per tree if 50 trees are planted per acre?

• 75 bushels

• 70 bushels

• 65 bushels

• 60 bushels

How does the yield per tree change with each additional tree planted?

• Increases by 3 bushels

• Decreases by 2 bushels

• Decreases by 3 bushels

• Increases by 2 bushels

What is the expression for the total number of trees per acre?

• 50 + X

• 65 - X

• 50 - X

• 65 + X

What is the simplified form of the harvest function H(x)?

• -3X^2 - 85X + 3250

• -3X^2 + 85X + 3250

• 3X^2 - 85X + 3250

• 3X^2 + 85X + 3250

What is the first derivative of the harvest function H(x)?

• -6X + 85

• -6X - 85

• 6X - 85

• 6X + 85

What is the critical number found by setting the first derivative to zero?

• X = 14

• X = -85

• X = 85

• X = -14

What does the second derivative test indicate about the function's concavity?

• Linear

• Concave down

• Constant

• Concave up

What is the ideal number of trees to plant per acre to maximize the harvest?

• 64 trees

• 36 trees

• 40 trees

• 50 trees

Why is the critical number rounded to the nearest whole number?

• Because X must be positive

• Because X represents a count of trees

• Because X is a fraction

• Because X is negative

What is the significance of the negative leading coefficient in the quadratic function?

• Indicates a constant function

• Indicates a linear function

• Indicates a maximum

• Indicates a minimum