2. Suppose f'(x) is a differentiable decreasing function for all x. In each of the following pairs, which number us the larger? Give a reason for your answer.
a. f'(5) and f'(6)
b. f''(5) and 0
c. f(5+△x) and f(5) + f'(5)△x
Differentiable Functions II
2007-10-12 11:37 pm
回答 (2)
2007-10-13 12:20 am
參考: My Maths knowledge
2007-10-13 12:19 am
(a)
Since f'(x) is decreasing, f'(5) > f'(6)
(b)
Since f'(x) is decreasing, d/dx (f'(x)) < 0 , i.e. f"(x) < 0
so f"(5) < 0
(c)
By mean value theorem.
f(5+△x) - f(5) = f'(c)△x, where c lies between 5 and 5+△x
case 1 △x > 0
Since f' is decreasing, so f'(c) < f'(5)
f(5+△x) - f(5) < f'(5)△x
i.e. f(5+△x) < f(5) + f'(5)△x
case 2 △x < 0
Since f' is decreasing, so f'(c) > f'(5)
f'(c)△x < f'(5)△x
f(5+△x) - f(5) < f'(5)△x (remember
i.e. f(5+△x) < f(5) + f'(5)△x
Since f'(x) is decreasing, f'(5) > f'(6)
(b)
Since f'(x) is decreasing, d/dx (f'(x)) < 0 , i.e. f"(x) < 0
so f"(5) < 0
(c)
By mean value theorem.
f(5+△x) - f(5) = f'(c)△x, where c lies between 5 and 5+△x
case 1 △x > 0
Since f' is decreasing, so f'(c) < f'(5)
f(5+△x) - f(5) < f'(5)△x
i.e. f(5+△x) < f(5) + f'(5)△x
case 2 △x < 0
Since f' is decreasing, so f'(c) > f'(5)
f'(c)△x < f'(5)△x
f(5+△x) - f(5) < f'(5)△x (remember
i.e. f(5+△x) < f(5) + f'(5)△x
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