Bounds Math Example 1
Follow the full solution, then compare it with the other examples linked below.
Example 1
mediumFor \(f(x) = -x^2 + 6x - 5\), find the maximum value (upper bound) on \([0, 5]\).
Solution
- 1 Find the vertex: \(x = -b/(2a) = -6/(2 \times -1) = 3\).
- 2 \(f(3) = -9 + 18 - 5 = 4\).
- 3 Check endpoints: \(f(0) = -5\), \(f(5) = -25+30-5=0\).
- 4 Maximum is \(f(3) = 4\).
Answer
Maximum value = 4 at \(x = 3\)
For a downward parabola, the vertex gives the maximum. The upper bound of \(f\) on \([0,5]\) is 4.
About Bounds
The upper and lower limits within which a quantity must lie; often expressed as .
Learn more about Bounds →More Bounds Examples
Example 2 hard
Show that for all real (x), (x^2 geq 0). What is the greatest lower bound (infimum) of (x^2)?
Example 3 mediumFor (g(x) = 3x + 1) on ([0, 4]), find the minimum and maximum values.
Example 4 hardProve that (sin(x) leq 1) and (sin(x) geq -1) for all (x) using the unit circle definition.