A function y = f(x) is convex (concave) if f''(x) = d2y/dx2 is positive (negative).
y = [c(1 - ) - 1] / (1 - )
y = [c(1 - ) / (1 - )] - [1 / (1 - )]
f'(c) = dy/dc = [1 / (1 - )] x [(1 - )] x c[(1 - ) - 1] = c (- )
f''(c) = d2y/dc2 = - x c(- - 1) = - / [c( + 1)]
Since 0 <= <= 2,
1 <= ( + 1) <= 3
Therefore, f''(c) > 0, signifying that y = [c(1 - ) - 1] / (1 - ) is a convex function.
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