đ Engineering Mathematics ·
đ Maxima & Minima of f(x,y) = x³y²(1−x−y)
A complete step-by-step solution to a repeated Anna University question from .
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đ Question
Find the maxima and minima of the function f(x,y) = x³y²(1 − x − y).
Step 1: Expand and differentiate
f(x,y) = x³y² − x⁴y² − x³y³
fâ = 3x²y² − 4x³y² − 3x²y³ = x²y²(3 − 4x − 3y)
f_y = 2x³y − 2x⁴y − 3x³y² = x³y(2 − 2x − 3y)
Step 2: Find the stationary points
Set fâ = 0 and f_y = 0. Besides the lines x = 0 and y = 0, we need:
4x + 3y = 3 and 2x + 3y = 2
Subtracting gives 2x = 1, so x = 1/2 and y = 1/3.
Key stationary point: (1/2, 1/3). The points on the axes (x = 0 or y = 0) are also stationary, but the second-derivative test is inconclusive there.
Step 3: Second-order derivatives
A = fxx = 6xy² − 12x²y² − 6xy³
B = fxy = 6x²y − 8x³y − 9x²y²
C = fyy = 2x³ − 2x⁴ − 6x³y
Step 4: Test the point (1/2, 1/3)
| A | B | C | AC − B² | Result |
|---|---|---|---|---|
| −1/9 | −1/12 | −1/8 | 1/144 > 0 | Maximum (A < 0) |
Step 5: Maximum value
f(1/2, 1/3) = (1/2)³ (1/3)² [1 − 1/2 − 1/3] = (1/8)(1/9)(1/6)
Maximum value = 1/432 at (1/2, 1/3). The function has no local minimum from this test.
đĄ Exam tip: the AC − B² test
- AC − B² > 0 and A < 0: maximum
- AC − B² > 0 and A > 0: minimum
- AC − B² < 0: saddle point
- AC − B² = 0: inconclusive
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