Observation is needed

Calculus Level 4

Let $$f$$ be a continuous and differentiable function in $$(x_1,x_2)$$. And the following conditions hold for it: $\begin{cases} f(x) \ f'(x) \geq x \ \sqrt{1-((f(x))^4} \\ \displaystyle \lim_{x \to x_1^+} ((f(x))^2=1 \\ \displaystyle \lim_{x \to x_2^-} ((f(x))^2=\frac{1}{2} \end{cases}$

Find the minimum value of $$(x_1^2-x_2^2)$$

Note : $$f'(x)=\dfrac{df(x)}{dx}$$

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