Bounds Math Example 3
Follow the full solution, then compare it with the other examples linked below.
Example 3
mediumFor \(g(x) = 3x + 1\) on \([0, 4]\), find the minimum and maximum values.
Solution
- 1 \(g\) is linear and increasing (slope = 3 > 0).
- 2 Minimum at left endpoint: \(g(0) = 1\).
- 3 Maximum at right endpoint: \(g(4) = 13\).
- 4 Range on \([0,4]\): \([1, 13]\).
Answer
Min = 1; Max = 13
A linear increasing function achieves its minimum at the left endpoint and maximum at the right endpoint of a closed interval.
About Bounds
The upper and lower limits within which a quantity must lie; often expressed as .
Learn more about Bounds →More Bounds Examples
Example 1 medium
For (f(x) = -x^2 + 6x - 5), find the maximum value (upper bound) on ([0, 5]).
Example 2 hardShow that for all real (x), (x^2 geq 0). What is the greatest lower bound (infimum) of (x^2)?
Example 4 hardProve that (sin(x) leq 1) and (sin(x) geq -1) for all (x) using the unit circle definition.