MathematicsChapter 2: Limit, Continuity and Derivative
Stationary Points and Optimization
Solve f'(x)=0 for candidates, classify them, and compare endpoints when the domain is closed.
Learning objectives
- State the rule for stationary points and optimization.
- Solve a stationary points and optimization problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Solve f'(x)=0 for candidates, classify them, and compare endpoints when the domain is closed.
Formula
f=x2-4x+7
Where / symbols
- Stationary points and optimization
- Solve f'(x)=0 for candidates, classify them, and compare endpoints when the domain is closed.
CONCEPT AND RULE
Solve f'(x)=0 for candidates, classify them, and compare endpoints when the domain is closed.
METHOD
For stationary points and optimization, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
For f=x2-4x+7, f'=2x-4=0 at x=2, giving minimum 3.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve stationary points and optimization
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- For f=x2-4x+7, f'=2x-4=0 at x=2, giving minimum 3.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Stationary points and optimization
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Stationary Points and Optimization
Sticky Notes · Formula and Worked Revision
Core Idea
- Solve f'(x)=0 for candidates, classify them, and compare endpoints when the domain is closed.
Rule / Formula
- Solve f'(x)=0 for candidates, classify them, and compare endpoints when the domain is closed.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For f=x2-4x+7, f'=2x-4=0 at x=2, giving minimum 3.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- Solve f'(x)=0 for candidates, classify them, and compare endpoints when the domain is closed.
- Worked pattern: For f=x^2-4x+7, f'=2x-4=0 at x=2, giving minimum 3.
- Final check: domain, sign, notation, units, and reasonableness.
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