Final answer:
The shortest distance Carl Friedrich Gauss comes from the origin while following the path f(x) = 4/x for 0.5 ≤ x ≤ 4 is approximately 4.442 units.
Step-by-step explanation:
To find the shortest distance between the origin and a point on the curve f(x) = 4/x, we'll use the distance formula D(x,y) = √(x² + y²). Here, we're interested in the distance from the origin (0,0) to a point (x, 4/x) on the curve.
The distance formula yields D(x) = √(x² + (4/x)²), representing the distance between the origin and the point on the curve. To find the minimum distance, we'll differentiate D(x) with respect to x and find its critical points.
D(x) = √(x² + (16/x²))
D'(x) = (1/2) * (x² + 16/x²)^(-1/2) * (2x - 32/x³)
Setting D'(x) = 0 to find critical points:
0 = 2x - 32/x³
2x = 32/x³
x⁴ = 16
x = (16)^(1/4)
x = 2 (ignoring the negative root as it's not within the given domain)
So, the critical point occurs at x = 2. To confirm this point yields the minimum distance, we'll evaluate D(x) at x = 2:
D(2) = √(2² + 4²) = √(4 + 16) = √20 ≈ 4.472
Therefore, the shortest distance Carl Friedrich Gauss comes from the origin is approximately 4.442 units when following the path f(x) = 4/x for 0.5 ≤ x ≤ 4.