Determine the Minimum Value of the Quadratic Expression (x^2 - x 1) Using Derivatives and Parabola Rules
Introduction
The expression (x^2 - x 1) is a quadratic polynomial. In this article, we'll use both calculus (derivatives) and the properties of parabolas to determine the minimum value of this quadratic expression. We will go through the process step by step to ensure a comprehensive understanding.
Derivatives Method
The given quadratic expression is (boldsymbol{x^2 - x 1}). We can also represent it as ((x^2 - 2x 1) 1 - 1 (x - 1)^2 1). Let's use derivatives to find its minimum value.
Find the derivative of the expression:The derivative of (x^2 - x 1) with respect to (x) is (frac{d}{dx}(x^2 - x 1) 2x - 1).
Set the derivative equal to zero and solve for (x):(2x - 1 0 Rightarrow x frac{1}{2})
Verify that this critical point is a minimum:Compute the second derivative, which is (frac{d^2}{dx^2}(x^2 - x 1) 2). Since the second derivative is positive, the function has a minimum at (x frac{1}{2}).
Calculate the value of the function at (x frac{1}{2}):(y left(frac{1}{2}right)^2 - frac{1}{2} 1 frac{1}{4} - frac{1}{2} 1 frac{3}{4})
Thus, the minimum value of the expression (x^2 - x 1) is (frac{3}{4}) at (x frac{1}{2}).
Parabola Rules Method
Another way to determine the minimum value of the quadratic expression is using the properties of parabolas.
Identify the form of the quadratic expression:The expression (x^2 - x 1) can be rewritten as ((x - frac{1}{2})^2 frac{3}{4}).
Find the vertex of the parabola:The vertex form of a parabola (y a(x - h)^2 k) has the vertex at ((h, k)). Here, (h frac{1}{2}) and (k frac{3}{4}). So, the vertex is at (left(frac{1}{2}, frac{3}{4}right)).
Determine the minimum value:Since a parabola opens upwards (the coefficient of (x^2) is positive), the minimum value of the expression occurs at the vertex. Therefore, the minimum value is (frac{3}{4}) at (x frac{1}{2}).
Hence, the minimum value of the quadratic expression (x^2 - x 1) is (frac{3}{4}).
Additional Insights
Graphical Interpretation: The graph of the function (y x^2 - x 1) is an upward-opening parabola. The minimum value of this parabola corresponds to the vertex of the parabola, which is also the point where the derivative is zero.
The Role of the Derivative: The first derivative (y' 2x - 1) helps identify the critical points, and the second derivative (y'' 2) confirms that the function is concave up, indicating a minimum.
Conclusion: Both methods, derivatives and the properties of parabolas, confirm that the minimum value of (x^2 - x 1) is (frac{3}{4}). This result can be verified by substituting (x frac{1}{2}) into the original expression.