How derivatives affect the shape of the graph

Question

For x in the interval (0,100), let f(x) = x100 +(100−x)100. Determine on what open intervals in (0, 100) the function f (x) is increasing and decreasing, and use this information to decide which of 33^100 + 67^100 or 41^100 + 59^100 is larger.

Answer 1

f(x) = x¹⁰⁰ + (100 - x)¹⁰⁰; f'(x) = 100x⁹⁹ - 100(100 - x)⁹⁹ = 0; x = 100 - x; x = 50; For 0 < x < 50 f'(x) < 0

and f(x) decreasing; for 50 < x < 100 f'(x) > 0 and f(x) increasing.

Because x = 33 < 50 and x = 41 < 50 f(x) decreasing we have that 33¹⁰⁰ + 67¹⁰⁰ > 41¹⁰⁰ + 59¹⁰⁰ is larger.