Example 4: unbounded region — is −300 the minimum?उदाहरण 4: अपरिबद्ध क्षेत्र — क्या −300 न्यूनतम है?Example 4: unbounded region — kya −300 minimum hai?

NCERT Fig 12.5 · Class 12 Maths · Chapter 12 · interactive diagram

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Parts and what they mean

Feasible region (unbounded) (सुसंगत क्षेत्र (अपरिबद्ध)) — It goes on forever towards the upper right, so it cannot be enclosed in a circle. A smallest corner value may NOT be the minimum here.
Open half-plane -50x + 20y < -300 (खुला अर्धतल -50x + 20y < -300) — i.e. −5x + 2y < −30 (red). It overlaps the feasible region, so Z goes below −300: Z has NO minimum.
Line -5x + 2y = -30 (रेखा -5x + 2y = -30) — The line Z = −300 drawn dashed (an open half-plane does not include its line).
Line 2x - y = -5 (रेखा 2x - y = -5) — Boundary of 2x − y ≥ −5; feasible side is below/right of it (O satisfies 0 ≥ −5).
Line 3x + y = 3 (रेखा 3x + y = 3) — Boundary of 3x + y ≥ 3; feasible side is away from O.
Line 2x - 3y = 12 (रेखा 2x - 3y = 12) — Boundary of 2x − 3y ≤ 12; feasible side is above it (towards O).
Corner (0, 5) (कोनीय बिंदु (0, 5)) — Z = −50(0) + 20(5) = 100 (the largest corner value; it IS the maximum — see the extra check).
Corner (0, 3) (कोनीय बिंदु (0, 3)) — Z = 20 × 3 = 60.
Corner (1, 0) (कोनीय बिंदु (1, 0)) — Z = −50 × 1 = −50.
Corner (6, 0): smallest corner value (कोनीय बिंदु (6, 0): सबसे छोटा कोनीय मान) — Z = −300. Smallest among corners, but not the minimum because the region is unbounded and the red half-plane meets it.
A common point (9, 3) (एक उभयनिष्ठ बिंदु (9, 3)) — Feasible (2·9 − 3·3 = 9 ≤ 12) and Z = −450 + 60 = −390 < −300. Proof that −300 is not the minimum.
x-axis (x-अक्ष) — The horizontal number line; inputs x are read here.
y-axis (y-अक्ष) — The vertical number line; outputs y = f(x) are read here.
Origin O(0, 0) (मूल बिंदु O(0, 0)) — Where the two axes cross.

Original diagram drawn by Simple Learning Hub, based on NCERT Fig 12.5. Report a mistake