How to use Lagrangian multipliers with inequality constraints?

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Can anyone assist me with guidance on how to solve the following max and min with constraints problem where the side conditions are inequalities, using Lagrangian multipliers? I was able to successfully solve the problem using other methods but, I have not had success in solving the problem using Lagrangian multipliers.

For any help you could provide, I will be greatly grateful beforehand. enter image description here