There is a minimum value of -81 located at (x, y) = (6, -3).
The function given to us is f(x, y) = 3y² - 3x².
The constraint given to us is 2x + y = 9.
Rearranging the constraint, we get:
2x + y = 9,
or, y = 9 - 2x.
Substituting this in the function, we get:
f(x, y) = 3y² - 3x²,
or, f(x) = 3(9 - 2x)² - 3x² = 3(81 - 36x + 4x²) - 3x² = 243 - 108x + 12x² - 3x² = 243 - 108x + 9x².
To find the extremum, we differentiate this, with respect to x, and equate that to 0.
f'(x) = - 108 + 18x ... (i)
Equating to 0, we get:
- 108 + 18x = 0,
or, 18x = 108,
or, x = 6.
Differentiating (i), with respect to x again, we get:
f''(x) = 18, which is greater than 0, showing f(x) is minimum at x = 6.
The value of y, when x = 6 is,
y = 9 - 2x,
or, y = 9 - 2*6 = 9 - 12 = -3.
The value of f(x, y) when (x, y) = (6, -3) is,
f(x, y) = 3y² - 3x²,
or, f(x, y) = 3*(-3)² - 3*6² = 3*9 - 3*36 = 27 - 108 = -81.
Thus, there is a minimum value of -81 located at (x, y) = (6, -3).
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