Maxima and minima
First and second derivative tests and simple applied problems.
The essentials
A function has a local maximum at c if f(c) is greater than or equal to the values of f at all nearby points, and a local minimum if it is less than or equal to them. At such a point of a differentiable function, f′(c) = 0; points where f′(c) = 0 (or f′ does not exist) are called critical points.
First derivative test: if f′ changes from positive to negative as x passes through c, f has a local maximum at c; from negative to positive, a local minimum; if the sign does not change, c is a point of inflection, not an extremum.
Second derivative test: if f′(c) = 0 and f″(c) < 0, c is a local maximum; if f″(c) > 0, a local minimum; if f″(c) = 0, the test fails and you go back to the first derivative test.
Key points
- Absolute (global) extrema of a continuous function on a closed interval [a, b] occur at critical points or at the end points a and b.
- To find them, evaluate f at every critical point inside the interval and at both end points, then compare.
- Completing the square gives quick answers for quadratics: x² + 4x + 9 = (x + 2)² + 5 has minimum 5.
- For two positive numbers with a fixed sum, the product is greatest when the numbers are equal.
Exam tips
- In word problems, write the quantity to optimise in terms of ONE variable using the given condition, then differentiate.
- State both where the extremum occurs and its value if asked.
Common mistakes
- Forgetting end points when finding absolute extrema on a closed interval.
- Giving the x-value when the question asks for the maximum value.