Application of Derivatives (Class 12 Maths): Maxima, Minima and Tangents Solved

Tangents and normals

Two special cases appear regularly: if f′(a) = 0 the tangent is horizontal, and if f′(a) is undefined the tangent is vertical, which makes the normal horizontal.

Finding maxima and minima

Critical points are where f′(x) = 0 or f′(x) does not exist. A critical point is only a candidate — it must then be classified.

1. First derivative test — if f′ changes from positive to negative, it is a local maximum; negative to positive, a local minimum; no sign change, neither.
2. Second derivative test — if f′(c) = 0 and f″(c) < 0 it is a local maximum; if f″(c) > 0 a local minimum. If f″(c) = 0 the test fails and you must return to the first derivative test.

The classic illustration is f(x) = x³ at x = 0: both f′ and f″ vanish, but there is no extremum at all — it is a point of inflection. That example is worth remembering precisely because it shows why the second derivative test cannot be applied blindly.

Absolute versus local extrema

On a closed interval [a, b], the absolute maximum and minimum may occur at a critical point or at an endpoint. Forgetting to evaluate the endpoints is the most common error in this section.

• Find all critical points inside the interval.
• Evaluate f at each critical point and at both endpoints a and b.
• The largest value is the absolute maximum, the smallest the absolute minimum.

FAQs

What is a critical point?

A point where the derivative is zero or undefined. It is a candidate for an extremum but must still be tested.

When does the second derivative test fail?

When f″(c) = 0. The point may be a maximum, a minimum or neither, so you must fall back on the first derivative test.

Is every critical point a maximum or minimum?

No. f(x) = x³ has f′(0) = 0 but no extremum at x = 0 — it is a point of inflection.

How do I find absolute extrema on a closed interval?

Evaluate the function at all interior critical points and at both endpoints, then compare the values.