Answer:
The minimum value of Q is 162
(x=3 and y=6)
Explanation:
To minimize the function Q = 6x^2 + 3y^2, subject to the constraint x + y = 9, we can graphically analyze the problem.
First, we rewrite the constraint equation x + y = 9 as y = 9 - x. Now, we substitute this expression for y into the objective function Q:
Q = 6x^2 + 3(9 - x)^2
Q = 6x^2 + 3(81 - 18x + x^2)
Q = 6x^2 + 243 - 54x + 3x^2
Q = 9x^2 - 54x + 243
Now, we can plot the graph of Q as a quadratic function of x. The vertex of this parabolic curve represents the minimum value of Q within the given constraint.
By completing the square or finding the vertex using the formula x = -b/(2a), we can determine that the x-coordinate of the vertex is x = 3. Substituting this value back into the constraint equation, we find that y = 9 - 3 = 6.
Therefore, the minimum value of Q occurs when x = 3 and y = 6, which corresponds to the point (3, 6) on the graph.
Hope it helps!! :)