# Find and indicate the minimum of a piecewise function in the Plot3D diagram

I'm trying to indicate in my Plot3D diagram the minimum level of the dependent variable and the corresponding levels of two independent variables. More specifically, $$i$$ is a function of $$k$$ and $$r$$, i.e. $$i=f(k,r;s,d,c,q)$$ where $$s$$, $$d$$, $$c$$, and $$q$$ are given constant. Depending on the ranges of $$k$$ and $$r$$, there are eight different functions of $$i$$ as follows:

i1 = (3 d k^2 r + k^2 r (k r - 3 s) + 3 d^2 s + c (6 d^2 (-1 + q) + 3 k^2 (-1 + 2 q) r + 6 d s))/(6 d s^2);
i2 = 1/(6 d (-1 + r)^2 s^2) (3 c (d^2 (-3 + q (2 - 4 r) + 5 r) + 2 d (k - 2 k r + k (-1 + 2 q) r^2 + s - r s) + k (k (-1 - r + 2 q r) (1 + (-3 + r) r) + 2 (-1 + r) r s)) + (-1 + r) (d^3 + 3 d^2 (k (-1 + r) - s) + 3 d k (k + k (-3 + r) r + 2 r s) + k^2 (k (-1 + r (3 + (-1 + r) r)) - 3 r^2 s)));
i3 = 1/(6 d (-1 + r) s^2) (d^3 + 3 d^2 (k (-1 + r) - s) + 3 d k (k + k (-3 + r) r + 2 r s) +
k^2 (k (-1 + r (3 + (-1 + r) r)) - 3 r^2 s) + 3 c (d^2 + 2 d k (-1 + 2 q) (-1 + r) - 2 d s +
k (k (-1 + 2 q (-1 + r)^2 + r - r^2) + 2 r s)));
i4 = (k^2 r (-3 d + 2 k r) - 3 (d - k r)^2 s - 3 c (2 d^2 (-1 + q) + k r (k - 2 s) + 2 d s))/(6 d (-1 + r) s^2);
i5 = (k^2 r (-3 d + 2 k r) - 3 (d - k r)^2 s - 3 c (2 d^2 (-1 + q) + k r (k - 2 s) + 2 d s))/(6 d (-1 + r) s^2);
i6 = (d^3 + 3 d^2 k r + 3 d k^2 (-1 + r) r + k^3 (-1 + r)^2 r + 3 c (d^2 (-1 + 2 q) + 2 d k r + k^2 (-1 + r) r))/(6 d r s^2);
i7 = - (d (d + c (3 + 6 (-1 + q) r)))/(6 (-1 + r) r s^2);
i8 = (d^3 + 3 d^2 k r + 3 d k^2 (-1 + r) r + k^3 (-1 + r)^2 r + 3 c (d^2 (-1 + 2 q) + 2 d k r + k^2 (-1 + r) r))/(6 d r s^2);


My goal is to find the minimum value of $$i$$ and the corresponding $$k$$ and $$r$$ for the following values of the constants:

s = 4; d = 0.9; c = 0.15; q = 1.5;


I used Piecewise and my Plot3D code is as follows:

Plot3D[Piecewise[{{i1, 0 <= k < d && 0 <= r <= 1}, {i2, d <= k < 2 d && 1/2 < r < (d/k)}, {i3, d <= k < 2 d && 1 - (d/k) < r <= 1/2}, {i4, d <= k < 2 d && 0 <= r <= 1 - (d/k)}, {i5, 2 d <= k <= s && 0 <= r < (d/k)}, {i6, d <= k < 2 d && (d/k) <= r <= 1}, {i7, 2 d <= k <= s && (d/k) <= r < 1/2}, {i8, 2 d <= k <= s && 1/2 <= r <= 1}}], {k, 0, s}, {r, 0, 1}, AxesLabel -> {k, r, i}]


which yields:

But, I would like to do two more things:

First, I would like to come up with concrete numbers, i.e. to find at which levels of $$k$$ and $$r$$, $$i$$ is minimized, and find that minimum value of $$i$$.

Second, I would like to indicate these in the diagram, i.e. the minimum $$i$$ and the corresponding levels of $$k$$ and $$r$$.

For the first task, I did the following without success:

Maximize[{Piecewise[{{i1, 0 <= k < d && 0 <= r <= 1}, {i2, d <= k < 2 d && 1/2 < r < (d/k)}, {i3,
d <= k < 2 d && 1 - (d/k) < r <= 1/2}, {i4, d <= k < 2 d && 0 <= r <= 1 - (d/k)}, {i5, 2 d <= k <= s && 0 <= r < (d/k)}, {i6, d <= k < 2 d && (d/k) <= r <= 1}, {i7, 2 d <= k <= s && (d/k) <= r < 1/2}, {i8, 2 d <= k <= s && 1/2 <= r <= 1}}] && 0 <= r <= 1 && 0 <= k <= 4}, {r, k}]


• As a plotting tip: you might want to add the setting Exclusions -> None if you are sure your function is supposed to be continuous at the breaks. – J. M.'s discontentment Apr 24 at 0:37