Respuesta :
Answer:
Given the statement: if y =3x+6.
Find the minimum value of [tex](x^3)(y)[/tex]
Let f(x) = [tex](x^3)(y)[/tex]
Substitute the value of y ;
[tex]f(x)=(x^3)(3x+6)[/tex]
Distribute the terms;
[tex]f(x)= 3x^4 + 6x^3[/tex]
The derivative value of f(x) with respect to x.
[tex]\frac{df}{dx} =\frac{d}{dx}(3x^4+6x^3)[/tex]
Using [tex]\frac{d}{dx}(x^n) = nx^{n-1}[/tex]
we have;
[tex]\frac{df}{dx} =(12x^3+18x^2)[/tex]
Set [tex]\frac{df}{dx} = 0[/tex]
then;
[tex](12x^3+18x^2) =0[/tex]
[tex]6x^2(2x + 3) = 0[/tex]
By zero product property;
[tex]6x^2=0[/tex] and 2x + 3 = 0
⇒ x=0 and x = [tex]-\frac{3}{2} = -1.5[/tex]
then;
at x = 0
f(0) = 0
and
x = -1.5
[tex]f(-1.5) = 3(-1.5)^4 + 6(-1.5)^3 = 15.1875-20.25 = -5.0625[/tex]
Hence the minimum value of [tex](x^3)(y)[/tex] is, -5.0625
