Q. Find the minimum value of Q=x^2y subject to the constraint 2x^2+4xy=294.
Its the derivative method.
Hints:
Here you have two variables, and one constraint. From the constraint, y can be solved in terms of x. This value of y can then be substituted into Q to make Q a function of Q=Q(x), whose minimum value can be calculated by differentiation.
I get x=±7√2.
Check each to make sure it is a minimum or maximum.
To find the minimum value of Q = x^2y, we need to use the derivative method to optimize the function subject to the given constraint.
Step 1: Write the given constraint equation in terms of one variable.
The constraint equation is 2x^2 + 4xy = 294. We can rewrite this equation to solve for y in terms of x: y = (294 - 2x^2) / 4x.
Step 2: Substitute the expression for y into the objective function.
Replace y in the objective function Q = x^2y with the expression we obtained from the constraint equation:
Q = x^2 * (294 - 2x^2) / 4x.
Simplifying this expression, Q = (147x - x^3) / 2.
Step 3: Take the derivative of the objective function with respect to x.
Differentiating Q = (147x - x^3) / 2 with respect to x, we get dQ/dx = (147 - 3x^2) / 2.
Step 4: Set the derivative equal to zero and solve for x.
To find the critical points, we set dQ/dx = 0 and solve for x:
(147 - 3x^2) / 2 = 0.
Simplifying, we get 147 - 3x^2 = 0.
Rearranging, we have 3x^2 = 147.
Dividing both sides by 3, we get x^2 = 49.
Taking the square root, we have x = ±7.
Step 5: Find the corresponding y values.
We substitute the x values back into the equation we obtained from the constraint equation:
For x = 7: y = (294 - 2(7^2)) / 4(7) = 21.
For x = -7: y = (294 - 2(-7^2)) / 4(-7) = -21.
Step 6: Evaluate the objective function at each critical point.
Substituting the corresponding (x, y) values into the objective function Q = x^2y:
For x = 7, y = 21: Q = (7^2)(21) = 1029.
For x = -7, y = -21: Q = (-7^2)(-21) = 1029.
Step 7: Compare the values of the objective function.
We see that both critical points produce the same minimum value of Q = 1029.
Therefore, the minimum value of Q=x^2y, subject to the constraint 2x^2+4xy=294, is 1029.