To find the minimum vertical distance between the two parabolas described by the equations y = x² + 1 and y = x + x², we first need to define the vertical distance between the two functions. This distance can be represented as:
d(x) = (x² + 1) – (x + x²) = 1 – x
Here, d(x) represents the vertical distance between the two parabolas at any point x.
Next, we want to minimize this distance. To do this, we first analyze the expression:
- As x increases, d(x) decreases linearly because the equation simplifies to d(x) = 1 – x.
- It’s clear that this linear function has a maximum point when x is smaller. The line will yield larger distances when closer to zero.
- Indeed, the minimum vertical distance will occur when x reaches its maximum value in its valid range.
Since we are analyzing the entire domain of real numbers, as x approaches positive infinity, the distance tends towards negative infinity, but we are interested in the minimum distance within a reasonable context.
Setting d(x) to zero to find critical points gives:
1 – x = 0 ⇒ x = 1
At this point, we can find the value of d(1):
d(1) = 1 – 1 = 0
Thus, the minimum vertical distance between the two given parabolas occurs at x = 1, and that distance is 0. At this point, both parabolas intersect, meaning there is no vertical distance between them. Therefore, the answer is:
Minimum vertical distance = 0