Convex Functions and Inequalities: Proving sqrt{1x} - sqrt{x} sqrt{1y} - sqrt{y} if and only if y x

Understanding the behavior of functions, particularly convex functions, is crucial in mathematics and optimization. In this article, we will delve into the properties of the square root function, specifically focusing on the inequality that sqrt{1x} - sqrt{x} sqrt{1y} - sqrt{y} if and only if x y. This result showcases the fundamental characteristics of convex functions and their implications in mathematical analysis.

What is Convexity?

A function is considered convex if a line segment joining any two points on its graph lies above or on the graph. Equivalently, a function is convex if its graph lies below any chord, and the second derivative of a twice-differentiable function is non-negative. Convexity is a key property that simplifies many optimization problems.

The Function at Hand: sqrt{1x} - sqrt{x}

Let's consider the function fx sqrt{1x} - sqrt{x}. Our goal is to prove that this function is negative and strictly decreasing, which will help us to establish the desired equality.

Key Feature of the sqrt{x} Function

The fundamental key feature of the function fx is its convexity. When we draw a line between two points on the curve sqrt{x}, the gradient of the line between these two points is shallower than the original curve. This property is crucial in proving the inequality we need.

Proof Using Differentiation

To prove that sqrt{1x} - sqrt{x} sqrt{1y} - sqrt{y} if and only if x y, we can use two different methods: differentiation and algebraic manipulation.

Differentiation Method

We start by differentiating the function fx sqrt{1x} - sqrt{x}.

$$frac{df}{dx} frac{1}{2sqrt{1x}} - frac{1}{2sqrt{x}}$$

Since 1/sqrt{1x} 1/sqrt{x} for all x 0, the derivative is clearly negative, indicating that the function is strictly decreasing.

Algebraic Manipulation Method

The alternative approach involves algebraic manipulation. An equivalent condition to the inequality is:

$$sqrt{1x}sqrt{y} - sqrt{x}sqrt{1y} 0$$

By squaring both sides, we can simplify the equation. Let's square both sides of the given condition:

$$sqrt{1x}sqrt{y} - sqrt{x}sqrt{1y} 0 Rightarrow 1xy - 2sqrt{xy} 1yx 0$$

After further simplification, we get:

$$xy x y - 2sqrt{xy} Rightarrow xy x y - 2sqrt{xy} Rightarrow (sqrt{x} - sqrt{y})^2 0$$

Since the square of a real number is zero if and only if the number itself is zero, we have:

$$sqrt{x} sqrt{y} Rightarrow x y$$

Conclusion

We have demonstrated that the function fx sqrt{1x} - sqrt{x} is strictly decreasing and convex. By using both differentiation and algebraic methods, we have shown that sqrt{1x} - sqrt{x} sqrt{1y} - sqrt{y} if and only if x y. This proof highlights the importance of understanding convex functions and their properties in mathematical analysis.

The last inequality sqrt{1x}sqrt{x} is increasing on [0, infty), which confirms that the function is strictly convex.

This result is significant in various fields, including optimization, economics, and applied mathematics. It provides a solid foundation for understanding the behavior of functions and solving related problems.