Results and Discussion

With the ultimate goal to find the main points of the system, meaning those points where his plane tangent in the surface has gradient value null. But first, we analyzed (using the Hessian matrix) on these important points to make possible to know whether these are the maximum points, minimum or saddle point of the function.

The calculation of the Hessian matrix is given by:

$$ H[z(x,y)] = \begin{bmatrix} \frac{\partial^2 z}{\partial x^2} & \frac{\partial^2 z}{\partial z,\partial y} \\ \frac{\partial^2 z}{\partial z,\partial y} & \frac{\partial^2 z}{\partial y^2} \end{bmatrix} $$

thus to find the value of the Hessian of z(x,y) we obtained the partial derivatives of the second order, whose values are below:

(3) $$ \frac{\partial^2 z}{\partial x^2} = 2c = - 0,08774 $$ (4) $$ \frac{\partial^2 z}{\partial y^2} = 2d = - 0,00178 $$ (5) $$ \frac{\partial^2 z}{\partial x,\partial y} = \frac{\partial^2 z}{\partial y,\partial x} = f = 0,00319 $$

with the known partial derivatives, we have the Hessian:

(6) $$ H[z(x,y)]=(-0,08774)(0,00178)-(0,00319)^2=-0,0001663533 $$

As H[f(x,y)]<0, we know that in this function there is a saddle point and there are no minimum or maximum values (Guidorizzi, 1997). Saddle point is understood as the point where the slope is zero, but it is not a maximum or minimum value. In it there is the highest elevation in one direction and lowest in the perpendicular direction.

Now, known the point we want to find, we take the partial derivatives of the first order function and made them equal to zero. To know:

(7) $$ \frac{dz}{dx}=a+2cx+fy=0 $$ (8) $$ \frac{dz}{dy}=b+2dy+fx=0 $$

Therefore:

(9) $$ -\frac{a+2cx}{f}=y=\frac{-b+fx}{2d} $$

replacing the values that are found in Table 2 of the appendix, we have that x=(2da-bf)/(f²-4cd)=1,5 and thus y=22.2 and z=0.63 . Then it is possible to concluded that for optimal absorbance point (z = 0.63), we know that it is necessary 22.2% PEG where it gets 1.5 weeks of plasmolysis.

Graph 3: Extrapolated fitted surface for extreme values of PEG concentrations and time..

Is possible to observe that for very long time values the absorbance decrease for zero very fast and the time do not pass longer than 8 weeks. The PEG concentrations also increase the absorbance values for both extreme concentrations, when approximate of zero and pass of 30% of PEG the absorbance value explodes.

Conclusions

Through these analyzes we could first find one function that suited in r² = 95%, which is very satisfactory, and validates our whole discussion about the obtained surface.

Then, through the Hessian we found a saddle point, which is also particularly relevant biologically. Because the absorbance measures the metabolic activity of bacteria, and how we want that they are plasmolysed, this value should be low, but not zero, because if so, they would be dead and would not serve for our propose. As it is a saddle point, we know that it is not null. Finally, the point was found through gradient of the function and we concluded that for the minimum value of absorbance the best PEG concentration is 22,2%.

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