Maxima and Minima
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Direct answer
Local extrema of a differentiable function occur only at critical points, where f'(x) = 0 or f' fails to exist, and they are classified either by the first derivative test — the sign change of f' across the point — or by the second derivative test: f''(c) > 0 gives a local minimum, f''(c) < 0 a local maximum, and f''(c) = 0 is inconclusive. On a closed interval [a, b], the absolute maximum and minimum are simply the best among the values of f at the critical points and the two endpoints.
What you must remember
- Second derivative test: f'(c) = 0 with f''(c) > 0 gives a local minimum, f''(c) < 0 a local maximum; f''(c) = 0 decides nothing — fall back to the first derivative test or higher derivatives.
- First derivative test: f' changing from + to - across c gives a local maximum, from - to + a local minimum, no sign change neither.
- Closed-interval method: list all critical points inside (a, b), evaluate f at each and at both endpoints, and take the largest and smallest values.
- Word problems: with a fixed sum, two positive numbers have maximum product when equal (AM-GM confirms); always respect the natural domain (lengths positive).
- Distance problems: to find the point on a curve nearest to a given point, minimise the square of the distance — the same minimiser with kinder algebra.
- Trigonometric shortcut: a sin x + b cos x ranges over [-sqrt(a^2 + b^2), sqrt(a^2 + b^2)], giving extrema without any calculus.
- Geometry of the tests: a minimum sits where the curve is concave up (holds water) and a maximum where it is concave down (spills water); f'' measures exactly this at the critical point.
Common confusion
Students equate f'(c) = 0 with "extremum at c". Both x^3 and x^4 have f'(0) = 0, yet x^3 has neither a maximum nor a minimum there while x^4 has a minimum — the sign chart of f' separates them. The second slip is forgetting endpoints in absolute-extrema questions: the absolute maximum can sit at an endpoint where f' never vanishes. Finally, a local maximum can lie far below an absolute maximum elsewhere; never answer a "greatest value" question with a local analysis alone.
Exam-focused takeaway
JEE Main asks direct second-derivative classification, absolute extrema on closed intervals and standard optimisation word problems — fences, boxes, cylinders inscribed in spheres — as numerical-value questions where clean algebra is the skill tested. JEE Advanced dresses the same machinery in parameters (extrema values as functions of a constant), functions defined by integrals, and existence arguments where monotonicity rather than computation locates the extremum. Whatever the form, the routine is fixed: domain, critical points, test, compare — in that order.
Frequently asked questions
What if f''(c) = 0 at a critical point?
It is silent; use the first derivative test — compare x^3 (neither) with x^4 (minimum) at 0.
Can the absolute maximum occur at an endpoint?
Yes. On a closed interval the candidates are the critical points and both endpoints; a monotonic function attains both extrema at the endpoints.
Which two numbers with a fixed sum have the greatest product?
Equal halves, by AM-GM or a one-line derivative argument.
How do I find the nearest point on a curve to an external point?
Minimise the squared distance rather than the distance; the minimiser is identical and the derivative is far simpler.
Is every critical point a local extremum?
No. Points where f' vanishes without a sign change, such as x = 0 for x^3, are stationary but not extrema.