Optimize for a parameter in a function with constraints on other 2 more parameters

I am an applied statistics student trying to solve a problem where constrained optimization is required. I have a function $$f(x, p_1, p_2)$$ in which $$p_1 \epsilon [0,1]$$, $$p_2 \epsilon [0,1]$$ are probabilities, and $$x \epsilon (-\infty,2)$$. I am looking for the optimal range of the 3rd parameter $$x$$ under the condition that $$f(x, p_1, p_2) > 0$$ only if $$p_1 \leq 0.9 \cap p_2 \leq 0.805$$. I naively calculated $$x$$ as the zero of the function $$f(x, p_1=0.9,p_2=0.805)$$. However, with some trial and error it quickly became clear to me that this leads to an incorrect solution. The function and what I tried in the R programming language is given below:

valueFunc = function(x, p1, p2){
b1 = (1-p1) * x + 2 - 4 * p2
b2 = (1-p1) * x + (1-p2) * x
a = 2-4*p1 + (p1-p2) * x

return(a - max(b1, b2))
}

#What I have tried
x=uniroot(valueFunc, interval = c(0,-500), p1=0.9, p2=0.805)$root  Please let me know how should I approach this problem to find the optimal range of $$x$$? Links to articles are welcome. A general direction and/or a solution from which I can learn is also welcome. Due to limited exposure to such problems I have probably not selected all tags correctly. • It would help if you can write out the objective function (e.g., you want to minimize$f(x,p_1,p_2)$for fixed$p_1$and$p_2$, or something like that), and what your constraints are (e.g.,$x \in (-\infty,2)\$.) It's a little hard to tell from what you wrote. – LarrySnyder610 May 7 at 15:49