If y = 2x - 8, what is the minimum value of the product xy?

Solution:

Let z = xy

Since y = 2x - 8

z = x (2x - 8)

z = 2x² - 8

Differentiate w.r.t to x,

dz /dx = 4x - 8

Differentiate again w.r.t z

d²z/dx² = 4

Now consider dz/dx = 0

⇒ 4x - 8 = 0

⇒ x = 2

When x = 2, d²z/dx² = 4 > 8

z is minimum when x = 2

Substitute x = 2 in y = 2x - 8

y = 2(2) - 8 = -4

Minimum value of z = xy

Value of z at x = 2 and y = -4

z = (2)(-4) = -8

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If y = 2x - 8, what is the minimum value of the product xy?

Summary:

If y = 2x - 8, the minimum value of the product xy is -8.