Bounds Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Bounds.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The upper and lower limits within which a quantity must lie; often expressed as a \leq x \leq b.

Temperature tomorrow will be between 60F and 75F. Those are bounds.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Bounds define the interval within which a value must lie; knowing bounds can solve problems even without exact values.

Common stuck point: Bounds can be strict (< and >) or inclusive (\leq and \geq).

Sense of Study hint: Substitute the boundary values into the expression to see if they are included (closed dot) or excluded (open dot).

Worked Examples

Example 1

medium
For \(f(x) = -x^2 + 6x - 5\), find the maximum value (upper bound) on \([0, 5]\).

Solution

  1. 1
    Find the vertex: \(x = -b/(2a) = -6/(2 \times -1) = 3\).
  2. 2
    \(f(3) = -9 + 18 - 5 = 4\).
  3. 3
    Check endpoints: \(f(0) = -5\), \(f(5) = -25+30-5=0\).
  4. 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.

Example 2

hard
Show that for all real \(x\), \(x^2 \geq 0\). What is the greatest lower bound (infimum) of \(x^2\)?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
For \(g(x) = 3x + 1\) on \([0, 4]\), find the minimum and maximum values.

Example 2

hard
Prove that \(\sin(x) \leq 1\) and \(\sin(x) \geq -1\) for all \(x\) using the unit circle definition.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

inequality intuition