Symbolic minimum value

I have the following polynomial expression v1[z]:

-((5 L^3 q z)/(12 J Y)) + (L^2 q z^2)/(4 J Y) - (L q z^3)/(12 J Y)


and I would like to symbolically evaluate its minimum in 0<z<L, with q>0, L>0, J>0 and Y>0. How can I do this?

I have tried the following numerical way:

minv1 = Minimize[{v1[z] /. q -> 1 /. L -> 1 /. J -> 1 /. Y -> 1,z > 0, z < 1}, z ]

• Since it is a cubic polynomial, it will have in general a local max and a local min, but no absolute. In any case, you can set the derivative to 0: Solve[D[-((5 L^3 q z)/(12 J Y)) + (L^2 q z^2)/(4 J Y) - (L q z^3)/(12 \ J Y), z] == 0, z], which gives {{z -> 1/3 (3 L - I Sqrt[6] L)}, {z -> 1/3 (3 L + I Sqrt[6] L)}}. Commented Dec 16, 2020 at 17:04
• For the numeric case given, minv1 = Minimize[{v[z] /. {q->1, L->1, J->1, Y->1}, 0<z<1}, z] gives a result on the boundary {-(1/4), {z -> 1}} Commented Dec 16, 2020 at 17:25

There is no global minimum:

Minimize[
{
(* expression *)
-((5 L^3 q z)/(12 J Y)) + (L^2 q z^2)/(4 J Y) - (L q z^3)/(12 J Y),
(* constraints *)
0 < z < L, q > 0, L > 0, J > 0, Y > 0
},
{z, q, L, J, Y}
]


Minimize::natt: The minimum is not attained at any point satisfying the given constraints.

{-∞, {z -> 1/2, q -> ∞, L -> 1, J -> 1, Y -> 1}}


The command of 12.2

ToRadicals[Minimize[{-((5 L^3 q z)/(12 J Y)) + (L^2 q z^2)/(4 J Y) - (L q z^3)/(12 J Y),
0 < z <= L && q > 0 && L > 0 && J > 0 && Y > 0}, z],
Assumptions -> q > 0 && L > 0 && J > 0 && Y > 0]


produces

$$\left\{ \begin{array}{cc} \{ & \begin{array}{cc} -\frac{L^4 q}{4 J Y} & Y>0\land q>0\land L>0\land J>0 \\ \infty & \text{True} \\ \end{array} \\ \end{array} ,\left\{z\to \begin{array}{cc} \{ & \begin{array}{cc} L & Y>0\land q>0\land L>0\land J>0 \\ \text{Indeterminate} & \text{True} \\ \end{array} \\ \end{array} \right\}\right\}$$

and the command

ToRadicals[Minimize[{-((5 L^3 q z)/(12 J Y))+(L^2 q z^2)/(4 J Y)-(L q z^3)/(12 J Y),0<z<L&&q>0&&L>0&&J>0&&Y>0},z],Assumptions->q>0&&L>0&&J>0&&Y>0]


performs

$$\left\{ \begin{array}{cc} \{ & \begin{array}{cc} -\frac{L^4 q}{4 J Y} & Y>0\land q>0\land L>0\land J>0 \\ \infty & \text{True} \\ \end{array} \\ \end{array} ,\{z\to \text{Indeterminate}\}\right\}$$

This means the minimum in z is not attained in the case of the strict inequalities z>0&&z<L.