Maxima & MinimaJEE Main

Concept

At a smooth peak or valley of a curve the tangent is horizontal, so the derivative is zero. These are the critical points. The second derivative then tells them apart: concave-down ($f''<0$) is a maximum, concave-up ($f''>0$) is a minimum.

Key formula

Derivation

For a cubic $f = ax^3+bx^2+cx+d$, the derivative $f'(x) = 3ax^2 + 2bx + c$ is a quadratic. Its discriminant decides how many turning points exist: two (distinct real roots), one inflection (repeated root), or none.

Evaluate $f''(x) = 6ax + 2b$ at each critical point to classify it. A cubic has at most one local max and one local min.

Scenarios to explore

• Maxima & Minima — Critical points of a cubic by the second-derivative test.

Real-world applications

• Optimisation: largest volume, least cost, shortest time.
• Finding equilibrium points where potential energy is stationary.
• Curve sketching for graph problems.

JEE exam tips

• If $f'$ has no real roots, the cubic is monotonic — no turning points at all.
• For closed intervals, the absolute extrema are among critical points and the two endpoints.

Common mistakes

• Stopping at $f'(x)=0$ without classifying with $f''$ or a sign chart.
• Confusing a local extremum with the global one on a closed interval — check endpoints too.
• Assuming $f''=0$ always means an inflection (need a sign change).

Exam traps to avoid

• A point with $f'=0$ and $f''=0$ may still be an extremum — fall back on the first-derivative sign test.
• Turning points need $f'$'s discriminant $> 0$; tune $c$ to make them appear or vanish.