If \( g(x)=f(x)+f(1-x) \) and \( f^{\prime \prime}(x)0 ; 0 \leq x \leq 1 \), show that \( g(x) \...

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If \( g(x)=f(x)+f(1-x) \) and \( f^{\prime \prime}(x)0 ; 0 \leq x \leq 1 \), show that \( g(x) \) increases in \( [0,1 / 2) \) and decreases in \( (1 / 2,1] \).
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