I need to export an image in high resolution in JPEG, JPG, and PDF formats. Right now, I'm using screenshots, which results in low clarity. Could you please help me with the correct syntax to generate a high-resolution, downloadable image in these formats? I'm attaching the file below:

q_new.mw

The two profit functions intersect at a certain point, but the graph is not clearly visible in the range of Cb​ from 30,000 to around 60,000. How can I adjust the plot to make this range more visible? What can i do such that two lines are seen distinct in that area?

Sheet:Q_12.mw

I'm working on an optimization problem involving a single decision variable p1, subject to four inequality constraints:

• Two upper bound constraints:

  p1 < b1,p1 < b2

• Two lower bound constraints:

  c1 ≤ p1,c2 ≤ p1

Effectively, the feasible region for p1 is:

max⁡(c1,c2)  ≤  p1  ≤  min⁡(b1,b2)

I have already formulated the Karush-Kuhn-Tucker (KKT) conditions for this setup, and now I'm trying to determine:

1. The optimal value p1∗​

2. The corresponding feasibility conditions

3. A case-wise breakdown depending on which constraints are active or inactive

Sheet:  Q_P1_Optimum_condition.mw

How can I rearrange the inequality to isolate p1​ on one side?
I'm currently using the 'isolate' function, but it's returning expressions with the signum function.

Q_isolate.mw

Given two optimality conditions defined as inequalities — a lower bound condition and an upper bound condition — under the assumptions that all parameters are positive and satisfy the relationships w>Pu>Ce>U[0]>0, and w>Pu>Ce,   and upsilon,varphi, tau0, eta are between (0,1), where 1>=varphi>=tau0>=0. and upsilon>eta

Under what conditions on the parameters does the lower bound condition become violated while the upper bound condition is also violated?

Detailed question in sheet attached: Q_Condition.mw

[ see also Q_Const.mw ]