## 3 Research Results and Discussion
### The Maximum of the Criterion Function--Distributions of the Random Variable of IFR and MTFR Classes  
In this part of the paper, we will describe the conditions sufficient for the existence of a maximum of the criterion function (10). The definition of sufficient conditions will be for two cases. The first concerns the class of distributions of a random variable, where the time to failure of a technical object, T\({}_{1}\), is a random variable with an increasing function of damage intensity, \(\lambda_{1}\)(t), i.e., T\({}_{1}\)\(\in\) IFR (Increasing Failure Rate). The second concerns the class of distributions of a random variable with a unimodal failure intensity function, i.e., T\({}_{1}\)\(\in\) MTFR (Mean Time to Failure or Repair). The random variable of the MTFR class and the results of research on the properties of variable distributions are presented in [20; 21; 22]. **Conclusion 2**. : _If time function \(T_{1}\in\) IFR, \(\lambda_{1}(t)\) is differentiable, \(\alpha<0\), \(\beta>0\), \(\gamma<0\), \(\beta+\gamma f_{1}(0^{+})>0\), \(\lambda_{1}(\infty)\)\(\alpha\)\(ET_{1}+\beta-\alpha<0\); then the criterion function \(g(x)\) reaches its maximum value._  
**Proof of Conclusion 2. **  
The derivative of the criterion function g(x) is of the form:  
\[g^{\prime}(x)=\{\alpha\;[f_{1}(x)\;ET_{1}(x)-R_{1}(x)\;F_{1}(x)]+\beta\;R_{1}( x)+\gamma\;f_{1}(x)]/M^{2}(x)\]  
where M(x) is the denominator of the criterion function, g(x). \(\square\)  
It is known that if the time to damage T\({}_{1}\) belongs to the class of distributions of the MTFR random variable, then the equality H(x) = \(\lambda_{1}(x)\)\(ET_{1}(x)-F_{1}(x)\geq 0\) for x \(\geq 0\) is true. The description of the classes of distributions of the MTHR random variable can be found in the scientific papers [20,21]. According to [20,21], some distributions of lifetimes with a unimodal failure intensity function belong to the MTFR class. Based on H\({}^{\prime}\)(x) = \(\lambda_{1^{\prime}}(x)\)\(ET_{1}(x)\), it can be said that, if the damage intensity function \(\lambda_{1}(t)\) increases, the function H(x) also increases. The class of distributions of a random variable with a non-decreasing damage intensity function (IFR) is contained in the MTFR class. The function and its derivative have the same sign. \[h(x)=\alpha\;[\lambda_{1}(x)\;ET_{1}(x)-F_{1}(x)]+\beta+\gamma\;\lambda_{1}(x)\]  
It is known that H(0\({}^{+}\)) = 0, hence h(0\({}^{+}\)) = \(\beta+\gamma\;f_{1}(0^{+})>0\). From the fact that \(\alpha<0\), \(\beta>0\), \(\gamma<0\) and the function H(x) increases, it follows that the function h(x) decreases from the value h(0\({}^{+}\)) = \(\beta+\gamma\;f_{1}(0^{+})>0\) to the value h(\(\infty)=\lambda_{1}(\infty)\)\(\alpha\)\(ET_{1}+\beta-\alpha<0\). It follows that the derivative of g'(x) changes sign exactly once from \"+\" to \"\(-\)\". From this, it can be concluded that the criterion function, g(x), reaches exactly one maximum. If \(\lambda_{1}(\infty)=\infty\), then the following conditions are sufficient for the criterion function g(x) to exist: T\({}_{1}\)\(\in\) IFR, differentiability \(\lambda_{1}(t)\), \(\alpha<0\), \(\beta>0\), \(\gamma<0\), \(\beta+\gamma\;f_{1}(0^{+})>0\). The Weibull distribution, with an increasing damage intensity function, is a good example of this. Based on conclusions 1 and 2, we can formulate another sufficient condition for the existence of a maximum of the criteria function:  
**Conclusion 3**. : _If time function \(T_{1}\in\) IFR, \(\lambda_{1}(t)\) is differentiable, \(\beta+\gamma\;\lambda_{1}(0^{+})>0\), \(p_{23}>\delta/ET_{3}\), \(\lambda_{1}(\infty)\)\(\alpha\)\(ET_{1}+\beta-\alpha<0\), then the criterion function \(g(x)\) reaches its maximum value._  
**Proof of Conclusion 3. **  
The existence of a maximum asymptotic availability coefficient can be written based on assumptions. On the basis of the criterion function, g(x), the availability coefficient is obtained assuming: z\({}_{1}\) = 1, z\({}_{2}\)  
**Conclusion 4**.