(T) or false (F). (1)If f \'(c) = 0, then f has a local maximum or minimum at c.
ID: 3285623 • Letter: #
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(T) or false (F). (1)If f '(c) = 0, then f has a local maximum or minimum at c. (2). If f has an absolute minimum value at c, then f '(c) = 0. (3). If f is continuous on (a, b), then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in (a, b). (4). If f is differentiable and f(-1) = f(1), then there is a number c such that | c |< 1 and f '(c) = 0. (5). If f '(x) < 0 for 1 < x < 6, then f is decreasing on (1, 6). (6). If f ''(2) = 0, then (2, f(2)) is an inflection point of the curve y = f(x). (7). If f' (x) = g '(x) for all 0 < x < 1, then f(x) = g(x) for 0 < x < 1.Explanation / Answer
1.) False, x = c could be a point of inflection a perfect example is
f(x) = x^3 at x = 0 f'(0) = 0 but x = 0 is not a min or a max.
2.) False, in close interval problems then the boundary points could be the min or max.
3.) False, because it's an open interval.
4.) True(this is Rolle's theorem)
5.) True
6.) False, you would need to know whether f(x) changes concavity about x = 2.
7.) False. example f(x) = x g(x) = x + 1
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