M​​inimizing related functions Complete each of the following parts.
b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?

Verified step by step guidance

1. First, recognize that the function ƒ(x) = (x - a₁)² + (x - a₂)² + (x - a₃)² is a sum of squares, which is a quadratic function in terms of x.
2. To find the value of x that minimizes this function, we need to take the derivative of ƒ(x) with respect to x. The derivative will help us find the critical points where the function could have a minimum.
3. Calculate the derivative: ƒ'(x) = 2(x - a₁) + 2(x - a₂) + 2(x - a₃). This simplifies to ƒ'(x) = 6x - 2(a₁ + a₂ + a₃).
4. Set the derivative ƒ'(x) equal to zero to find the critical points: 6x - 2(a₁ + a₂ + a₃) = 0.
5. Solve for x: x = (a₁ + a₂ + a₃) / 3. This value of x minimizes the function ƒ(x) because it is the point where the derivative is zero, indicating a potential minimum.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. They graph as parabolas, which can open upwards or downwards depending on the sign of the coefficient 'a'. The vertex of the parabola represents the minimum or maximum point of the function, which is crucial for optimization problems.

Minimization

Minimization in calculus involves finding the lowest point of a function, which can be achieved using techniques such as taking the derivative and setting it to zero to find critical points. The second derivative test can then confirm whether these points are minima or maxima. In the context of the given function, we seek the value of x that minimizes the sum of squared differences.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making it easier to analyze its properties. This technique is particularly useful for finding the vertex of a parabola, which directly relates to identifying the minimum value of a quadratic function. In the context of the problem, it can help simplify the expression to find the optimal x value.