Calculus
Class 12

Maxima and Minima Conditions

f(x)=0 and f(x)<0maximumf'(x) = 0 \text{ and } f''(x) < 0 \Rightarrow \text{maximum}
−202−30−20−100102030
f'(x)=0, f''(x)<0 → maximum; f''(x)>0 → minimum

Adjust Variables

Coefficient a
a =
-22

Critical points occur where derivative is zero. Second derivative test determines if it's a maximum or minimum.

Real-World Applications

Business — Maximizing profit: Set d(Profit)/dx = 0.

Engineering — Optimizing structural weight with strength constraints.

Physics — Finding equilibrium positions.

Economics — Minimizing cost or maximizing utility.

Manufacturing — Optimal batch sizes.

Packaging — Minimizing surface area for given volume.

Transportation — Shortest route problems.

Agriculture — Maximizing crop yield with fertilizer.

Medicine — Optimal drug dosage.

Finance — Portfolio optimization.

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