- Represent the quadratic regression as \(f(X) = \beta_2 X^2 + \beta_1 X + \beta_0\).
- Calculate the first derivative: \(f'(X) = 2\beta_2 X + \beta_1\).
- Set \(f'(X) = 0\) and solve for \(X\): \(2\beta_2 X = -\beta_1 \implies X = -\frac{\beta_1}{2\beta_2}\).
- The turning point occurs at \(X = \boxed{-\frac{\beta_1}{2\beta_2}}\).
Explanation
- Problem Analysis
The quadratic regression model is defined by the equation \(Y = \beta_0 + \beta_1 X + \beta_2 X^2 + u\). To find the turning point (where the function reaches its local maximum or minimum), we treat the deterministic part of the model as a quadratic function of \(X\):
\[f(X) = \beta_2 X^2 + \beta_1 X + \beta_0\]
The turning point of a quadratic function occurs at the value of \(X\) where its slope (the first derivative) is zero.
- Finding the First Derivative
To find the derivative of \(f(X) = \beta_2 X^2 + \beta_1 X + \beta_0\) with respect to \(X\), we apply the power rule, which states that \(\frac{d}{dX}(aX^n) = naX^{n-1}\).
The derivative is:
\[f'(X) = \frac{d}{dX}(\beta_2 X^2) + \frac{d}{dX}(\beta_1 X) + \frac{d}{dX}(\beta_0)\]
\[f'(X) = 2\beta_2 X + \beta_1\]
- Solving for the Turning Point X
We set the first derivative equal to zero to find the critical point:
\[2\beta_2 X + \beta_1 = 0\]
Subtracting \(\beta_1\) from both sides gives:
\[2\beta_2 X = -\beta_1\]
Dividing by \(2\beta_2\) (assuming \(\beta_2
eq 0\)) yields the expression for the turning point:
\[X = -\frac{\beta_1}{2\beta_2}\]
- Conclusion
Comparing the result \(X = -\frac{\beta_1}{2\beta_2}\) with the given options, we see that it matches both option (a) and option (b). Therefore, the turning point occurs at \(X = -\frac{\beta_1}{2\beta_2}\).
Examples
In economics, a quadratic regression might model the relationship between a firm's output and its total cost, where \(Y\) represents cost and \(X\) represents units of output. Finding the turning point is essential for determining the output level that minimizes the average cost or optimizes production efficiency. By calculating the derivative of the cost function and setting it to zero, an economist can identify the precise level of production that achieves this goal, using the formula \(X = -\frac{\beta_1}{2\beta_2}\).