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?

1 Answer
Apr 19, 2018

#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#