What you'll learn
Using derivatives to analyze a function's shape and solve optimization problems.
First derivative
- f′ > 0 → increasing; f′ < 0 → decreasing.
- Critical points where f′ = 0 or undefined. First derivative test: sign change +→− = local max; −→+ = local min.
Second derivative
- f″ > 0 → concave up; f″ < 0 → concave down.
- Inflection point where concavity changes (f″ changes sign).
- Second derivative test: at a critical point, f″ > 0 → min; f″ < 0 → max.
Key theorems
- Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), some c has f′(c) = (f(b)−f(a))/(b−a).
- Extreme Value Theorem: a continuous function on a closed interval attains a max and min.
Optimization
Write the quantity to optimize, reduce to one variable using a constraint, differentiate, find critical points, and verify max/min (check endpoints too).
Exam tips
- Justify max/min with a sign analysis, not just 'f′=0'.
- Always check endpoints on closed intervals.
Common mistakes
- Calling every critical point an extremum (check the sign change).
- Forgetting to verify with the first/second derivative test.