Question

If `y=3x+6`, what is the minimum value of `x^3y`?

I don't even know what the question is asking... Since it's a linear function, is there a minimum value?

Answer 1

`f(-3/2) = -81/16`

Explanation:

The question gives us a two variable function:

`f(x,y) = x^3y" [1]"`

and askes us to find the minimum where:

`y = 3x+6" [2]"`

We can turn equation [1] into a single variable function by substituting equation [2] into it.

`f(x) = x^3(3x+6)`

Simplify:

`f(x) = 3x^4+6x^3`

graph{3x^4+6x^3 [-11.25, 11.25, -5.63, 5.62]}

Compute the first derivative:

`f'(x) =12x^3+18x^2`

We know that the extrema occur at zeros:

`0 =12x^3+18x^2`

`x^2(12x+18) = 0`

`x = 0` and `x = -3/2`

Use the second derivative test:

`f''(x) = 36x^2+36x`

`f''(0) = 0 larr` inflection point.

`f''(-3/2) = 36(-3/2)^2+36(-3/2) = 27 larr` local minimum

`f(-3/2) = 3(-3/2)^4+ 6(-3/2)^3`

`f(-3/2) = -81/16`