Final answer:
The largest t-interval where a unique solution is assured is found by applying relevant theorems that consider the functions' continuity and differentiability. Without specific details, one can usually use statistical tools like a TI calculator to work with t-distributions for probabilities and confidence intervals.
Step-by-step explanation:
The largest t-interval on which a unique solution is guaranteed to exist is typically determined by considering the behavior of the differential equations involved and their initial conditions, applying theorems such as the Picard-Lindelöf theorem, which assures a unique solution given certain conditions on the functions involved. The context provided suggests we are examining a statistical or probabilistic model, possibly related to confidence intervals or hypothesis testing where t-distributions are relevant.
Tools like the TI-83, 83+, and 84+ calculators can be used to find probabilities and inverse probabilities associated with the t-distributions using functions like tcdf and inverse t-probabilities. Without a specific differential equation or more details about the figure or the equation referenced, we can't provide the precise interval. However, in general, the largest interval where a unique solution can be guaranteed will be determined by the region where the function behaves 'nicely' - in a continuous and differentiable manner, and where any conditions required by the relevant theorems are met.