Consider the function \( f(x)=\max \left\{x^{2}\right. \), \( \left.(1-x)^{2}, 2 x(1-x)\right\},...

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Consider the function \( f(x)=\max \left\{x^{2}\right. \),
\( \left.(1-x)^{2}, 2 x(1-x)\right\}, x \in[0,1] \)
Let RMVT is applicable for \( f(x) \) on \( (a, b) \), then \( a+b+c \) is (where \( c \) is the point such that \( f^{\prime}(c)=0 \) )
(a) \( \frac{2}{3} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{3}{2} \)
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