How to Find the Minimum of a Quadratic Function

Understanding how to find the minimum value of a quadratic function is a fundamental concept in mathematics and optimization. This guide will walk you through the process, providing a clear explanation that is both comprehensive and easy to understand, making it easy for SEO optimization on Google.

Introduction to Quadratic Functions

A quadratic function is generally represented in the form of (f(x) ax^2 bx c), where (a), (b), and (c) are constants, and (a eq 0). These functions form a parabola shape in a coordinate system, and they can be approached from different angles—such as algebraically or geometrically. In this article, we will focus primarily on the algebraic approach using calculus and completing the square.

Using Calculus to Find the Minimum

One effective way to find the minimum value of a quadratic function is to use the technique of differentiation. Let's take a look at a specific form of a quadratic function in matrix notation:

For a quadratic form written as (frac{1}{2}x^T Qx - b^T x), where (Q) is a positive definite matrix, the function will have a unique minimum. If you want to understand this further, you can refer to the discussion in the comments, but we will proceed directly with solving the minimum.

Deriving the Minimum Using the Quadratic Formula

Consider the function:

(f(x) frac{1}{2} x^T Qx - b^T x frac{1}{2} b^T Q^{-1} b), where (Q^{-1} b) is chosen such that the function is minimized.

To find the minimum, we can complete the square. This means transforming the expression into a form that makes the minimum clear. Let's start from the matrix form:

(frac{1}{2}x^T Qx - b^T x frac{1}{2} b^T Q^{-1} b frac{1}{2}(x - a)^T Q(x - a) - frac{1}{2}a^T Qa frac{1}{2} b^T Q^{-1} b)

Here, (a Q^{-1} b). Thus, the expression simplifies to:

(f(x) frac{1}{2}(x - a)^T Q(x - a) frac{1}{2} b^T Q^{-1} b - frac{1}{2}a^T Qa frac{1}{2}(x - a)^T Q(x - a) C), where (C frac{1}{2} b^T Q^{-1} b - frac{1}{2}a^T Qa) is a constant.

The term (frac{1}{2}(x - a)^T Q(x - a)) is minimized when (x a Q^{-1} b). Therefore, the minimum value of the function is:

(C frac{1}{2} b^T Q^{-1} b - frac{1}{2}a^T Qa frac{1}{2} b^T Q^{-1} b - frac{1}{2} (Q^{-1} b)^T Q Q^{-1} b frac{1}{2} b^T Q^{-1} b - frac{1}{2} b^T Q^{-1} b 0).

Hence, the minimum value of the function is:

(frac{1}{2} b^T Q^{-1} b).

Using Algebra to Find the Minimum

Alternatively, you can also find the minimum of a simpler quadratic function using algebra. Consider a quadratic function in a standard form:

(f(x) ax^2 bx c)

If (a > 0), the function represents a parabola that opens upwards and has a minimum value. The minimum value can be found by completing the square or by using the vertex formula. The vertex formula for the x-coordinate is:

(x -frac{b}{2a})

Substitute this value of (x) back into the function to find the minimum value:

(fleft(-frac{b}{2a}right) aleft(-frac{b}{2a}right)^2 bleft(-frac{b}{2a}right) c frac{b^2}{4a} - frac{b^2}{2a} c c - frac{b^2}{4a})

Thus, the minimum value of the function is:

(c - frac{b^2}{4a}).

Conclusion

In summary, to find the minimum value of a quadratic function, you can use either the calculus method of differentiation or the algebraic method of completing the square. Both approaches provide a clear understanding of the process and help in optimizing your search results on Google.

Keywords: quadratic function, minimum value, optimization