Show that if 0 ≤a ≤b, then √(a) ≤√(b)

Name: Problem 51, Chapter 1: Calculus: One Variable
Uploaded: 2021-07-28T14:51:00-08:00
Duration: 1 min 9 s
Channel: Carson Merrill
Description: Show that if 0 ≤a ≤b, then √(a) ≤√(b)

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Key Concepts

Ordering of Real Numbers

This concept involves the structured comparison between numbers. When we say that 0 ? a ? b, we are using the ordering properties of the real numbers to compare values, which is essential in forming relationships and ensuring that operations applied to these numbers, such as taking a square root, respect this order.

Square Root Function

The square root function is defined for nonnegative numbers and returns the unique nonnegative number whose square equals the original number. Its key property is that it is well-behaved (monotonic) over its domain, which allows us to relate the order of the inputs to the order of their outputs.

Monotonicity of Functions

Monotonic functions, especially monotonically increasing functions, have the property that if one input is less than or equal to another, the output will maintain that order when the function is applied. This principle is crucial in the proof that if 0 ? a ? b, then applying the square root function—a monotonically increasing function—yields sqrt(a) ? sqrt(b).

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