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(reminder: all quotes here are fiddled, probably.)


Some Thoughts on Examinations (WW Sawyer)

2) Give examples of continuous functions that satisfy the following conditions, or explain why the requirements are impossible.
(a) f (0) = 1, f (1) =0, f'(x) positive for all x.
(b) f (0) = 1, f' (0) = -1, f"(x) always negative, f(x) =0 having no real solution.

Of course, in each part of this question the requirements are impossible. In (a) the function is required to be increasing and yet have f(1) less than f(0). In (b) the graph must have the slope and curvature as shown here, which clearly compels f (x) =0 to have a solution between x=0 and x=l, contrary to the last requirement.



:: Blogger h (19.01.10, 06:37   ) sagt...

"In (a) the function is required to be increasing "

that is a little vague.
what do you mean by "required to be increasing"? you are given info on the derivative, not on the function itself.

to make that passage from local (f'(x)) to global information (f(x)), one would need to use the intermediate value theorem.


:: Blogger sf (21.01.10, 00:40   ) sagt...


謝謝你補充了, 要進一步證明那一點, 關鍵何在.



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