Maxima & Minima – IIT‑JEE (Main + Advanced)
(Mathematics – Calculus – Section 3)
Goal – Quickly identify the local and global extrema of a function (single‑variable or multivariable) and solve constrained optimisation problems that appear in JEE‑Main/Advanced.
1. Simple Explanation
| Concept | What it means | Why it matters for JEE |
|---|---|---|
| Local maxima/minima | A point where the function takes a larger (or smaller) value than all nearby points. | Many exam questions ask for local extremes of a given function. |
| Global (absolute) maxima/minima | Largest/smallest value over the whole domain (or a closed interval). | For closed bounded regions (e.g., on a circle or rectangle) the global extreme always exists. |
| Critical point | Point where the derivative (or gradient) is zero or undefined. | All local extremes lie at critical points or on the boundary. |
| First‑derivative test | Look at the sign of (f') (or (\nabla f)) before and after a critical point. | Quick way to decide whether a critical point is a max, min or neither. |
| Second‑derivative test | Use (f'') (or Hessian) at a critical point. | Gives a definitive answer (except when (f''=0)). |
| Lagrange multipliers | Technique to find extrema of (f(x,y)) subject to (g(x,y)=c). | Classic JEE tool for constrained optimisation. |
| Boundary check | Evaluate (f) on the boundary of the domain. | Needed to find global extremes on closed regions. |
2. Key Points (Bullet‑style)
2.1 One‑Variable Functions
| Step | What to Do | Key Formula |
|---|---|---|
| 1 | Find critical points: (f'(x)=0) or (f') undefined. | (f'(x)=0) |
| 2 | First‑derivative test: • If (f') changes from + to – → local max. • If – to + → local min. |
- |
| 3 | Second‑derivative test: • If (f''(x_c)>0) → local min. • If (f''(x_c)<0) → local max. • If (f''=0) → inconclusive. |
(f''(x_c)) |
| 4 | For a closed interval ([a,b]), evaluate (f(a),f(b)) and compare with interior extrema to get global max/min. | - |
2.2 Two‑Variable Functions (Unconstrained)
| Step | What to Do | Formula |
|---|---|---|
| 1 | Find critical points: (\nabla f = (f_x, f_y) = (0,0)). | (\frac{\partial f}{\partial x} = 0,\ \frac{\partial f}{\partial y}=0) |
| 2 | Compute Hessian determinant (D = f_{xx}f_{yy} - (f_{xy})^2). | (D=f_{xx}f_{yy}-(f_{xy})^2) |
| 3 | Apply second‑derivative test: • (D>0) and (f_{xx}>0) → local min. • (D>0) and (f_{xx}<0) → local max. • (D<0) → saddle point. • (D=0) → inconclusive. |
- |
| 4 | For a closed bounded domain, also evaluate (f) on the boundary (via parametrisation or Lagrange). | - |
2.3 Constrained Optimisation (Lagrange Multipliers)
| Step | What to Do | Formula |
|---|---|---|
| 1 | Set up Lagrangian: (\mathcal{L}(x,y,\lambda)=f(x,y)-\lambda(g(x,y)-c)). | (\mathcal{L}=f-\lambda(g-c)) |
| 2 | Solve (\nabla_{x,y,\lambda}\mathcal{L}=0): • (f_x = \lambda g_x) • (f_y = \lambda g_y) • (g(x,y)=c). |
System of equations |
| 3 | Find all real solutions ((x,y,\lambda)). | - |