Is there an example of a function that is concave down on (0,1) with no local extrema on (0,1)?

Question

Also, why is the given example sufficient?

Answer 1

f(x) = -2x² + 8x + 10 is one such example

f'(x) = -4x + 8 f'(0)=8, f'(1)=4, f'(2) = 0 (critical point is at x=2, outside (0,1))

f''(x) = -4 (concave down for all x)