Find a function h(x) that represents the center of the largest inscribable circle between two given functions f(x) and g(x) | Step-by-Step Solution

What We're Solving:

We need to find a function h(x) that gives us the y-coordinate of the center of the largest possible circle that can fit between two given functions f(x) and g(x) at any point x. Think of it like finding the "middle path" that's equidistant from both curves!

The Approach:

The center of the largest inscribed circle must be equidistant from both boundaries. This means we're looking for points that are exactly halfway between f(x) and g(x), but we also need to ensure our circle actually fits without "poking through" either function.

The key insight is that the largest circle at any x-position will have its center on the line equidistant from both functions, and its radius will be half the distance between the functions at that point.

Step-by-Step Solution:

Step 1: Understand what "enclosed area" means First, we need to identify which function is on top and which is on bottom. Let's assume g(x) ≥ f(x) in our region of interest (g is above f).

Step 2: Find the midpoint between the functions The center of our largest circle must lie on the line that's exactly halfway between the two functions:

• This midpoint line is: h(x) = [f(x) + g(x)]/2

Step 3: Verify this gives us the largest possible circle At any point x, if our circle's center is at (x, h(x)), then:

• Distance to the upper boundary g(x) = g(x) - h(x) = g(x) - [f(x) + g(x)]/2 = [g(x) - f(x)]/2
• Distance to the lower boundary f(x) = h(x) - f(x) = [f(x) + g(x)]/2 - f(x) = [g(x) - f(x)]/2

Perfect! Both distances are equal, confirming our circle is centered properly.

Step 4: Confirm this is indeed the largest possible circle Any circle with center above or below h(x) would be closer to one boundary than the other, limiting its maximum radius. Our approach gives us a radius of [g(x) - f(x)]/2, which is the maximum possible.

The Answer:

h(x) = [f(x) + g(x)]/2

This elegant result tells us that the center of the largest inscribable circle is always at the arithmetic mean of the two bounding functions!

Memory Tip:

Think of it as "splitting the difference" - just like when you're trying to park exactly in the middle of two cars, you want to be equidistant from both boundaries. The math formalization of "right in the middle" is simply the average of the two functions! 🎯

Great question - this type of optimization problem beautifully combines geometric intuition with algebraic precision!