What is the minimum value of the product xy if y = 2x + 8?

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Finding the Minimum Value of the Product xy

To determine the minimum value of the product xy given the equation y = 2x + 8, we can start by rewriting the product in terms of a single variable.

First, replace y in the product:

xy = x(2x + 8) = 2x^2 + 8x

Now, we need to minimize the quadratic expression f(x) = 2x^2 + 8x. This is a standard quadratic function, and it has the general form of ax^2 + bx + c, where a = 2, b = 8, and c = 0.

The vertex of a parabola defined by a quadratic function ax^2 + bx + c gives the minimum (or maximum) value of the product when a > 0. The x-coordinate of the vertex can be found using the formula:

x = -b/(2a)

Plugging in the values:

x = -8/(2 * 2) = -8/4 = -2

Next, substitute x = -2 back into the equation for y to find the corresponding y value:

y = 2(-2) + 8 = -4 + 8 = 4

With these values, we can now calculate the product xy:

xy = (-2)(4) = -8

Therefore, the minimum value of the product xy is -8.

Conclusion

In summary, by substituting y = 2x + 8 into the product and finding the vertex of the resulting quadratic expression, we conclude that the minimum value of xy is -8 when x = -2 and y = 4.

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