- The quadratic function is defined as \(f(X) = \beta_0 + \beta_1 X + \beta_2 X^2\).
- The derivative is calculated as \(f'(X) = \beta_1 + 2\beta_2 X\).
- Setting the derivative to zero and solving for \(X\) yields the turning point \(X = -\frac{\beta_1}{2\beta_2}\).
- This matches option a, giving the final answer \(\boxed{a}\).
Explanation
- Problem Analysis and Function Identification
The quadratic regression function is given by \(Y = \beta_0 + \beta_1 X + \beta_2 X^2 + u\). To find the turning point of this function, we treat it as a deterministic function \(f(X) = \beta_0 + \beta_1 X + \beta_2 X^2\) by ignoring the error term \(u\). A turning point of a smooth function occurs where its first derivative with respect to the independent variable \(X\) is equal to zero.
- Finding the First Derivative
To find the derivative of \(f(X) = \beta_0 + \beta_1 X + \beta_2 X^2\) with respect to \(X\), we apply the power rule to each term individually:
\(f'(X) = \frac{d}{dX}(\beta_0) + \frac{d}{dX}(\beta_1 X) + \frac{d}{dX}(\beta_2 X^2)\)
Since \(\beta_0, \beta_1, \beta_2\) are constants, this simplifies to:
\(f'(X) = 0 + \beta_1 + 2\beta_2 X = \beta_1 + 2\beta_2 X\)
- Solving for the Turning Point
A turning point occurs at the value of \(X\) where the derivative is zero. Setting the derivative equal to zero, we have:
\(\beta_1 + 2\beta_2 X = 0\)
Subtract \(\beta_1\) from both sides:
\(2\beta_2 X = -\beta_1\)
Dividing both sides by \(2\beta_2\) (assuming \(\beta_2
eq 0\)), we find the location of the turning point:
\(X = -\frac{\beta_1}{2\beta_2}\)
- Conclusion and Selection of the Correct Option
Comparing our derived result \(X = -\frac{\beta_1}{2\beta_2}\) with the given options, we see that it corresponds to options a and b. Thus, the turning point occurs at \(X = -\frac{\beta_1}{2\beta_2}\). The final answer is \(\boxed{a}\).
Examples
Quadratic regression is widely used in economics and business to model relationships where the effect of a variable changes direction. For instance, consider the law of diminishing marginal returns, where an increase in production inputs initially boosts output, but eventually causes it to decline due to inefficiencies. By fitting a quadratic model to production data, an analyst can calculate the specific input level \(X = -\frac{\beta_1}{2\beta_2}\) that maximizes total output, allowing companies to operate at their most efficient point.