Animated Solution for Mathematics - Differentiation: Let $x$ and $y$ be two real variables such that $x>0$ and $xy=1$. Find the minimum value of $x+y$.

We are operating in the first quadrant of the Cartesian plane, constrained by the rectangular hyperbola defined by the equation:

We aim to minimize the sum $S=x+y$. Geometrically, this is equivalent to finding the smallest constant $k$ such that the line $x+y=k$ intersects the hyperbola.

Bridging Variables

Since $xy=1$ and $x>0$, we can express $y$ as $y=\frac{1}{x}$. This allows us to define the sum as a single-variable function:

To find the critical points, we differentiate $S(x)$ with respect to $x$:

Setting the derivative to zero to find the stationary point:

Given the constraint $x>0$, we discard the negative root and conclude that the critical point occurs at $x=1$.

Confirming the Minimum

To verify that this critical point is a minimum, we examine the second derivative:

Evaluating this at our critical point $x=1$:

Since the second derivative is positive, the function is concave up at $x=1$, confirming a local minimum. Substituting $x=1$ back into the sum, we find:

AM-GM Inequality

For any positive real numbers $x$ and $y$, the Arithmetic Mean-Geometric Mean (AM-GM) inequality states:

Substituting our constraint $xy=1$ into the inequality:

This confirms that the minimum value of the sum is 2, occurring precisely when $x=y=1$.