请问一道10年的题

zhouxiaofei170UID: 3185081

If f is strictly increasing, then which of the following is necessarily WRONG?
C integral of f(x) from 0 to 1 = integral of f(x) from 1 to 2
E f'(1)=-f'(2)

怎么觉得这两个都是错的？

请指教。

songfjUID: 3089927

同问。

我也感觉两个都是错的呀。

zzb198484UID: 3286768

Does "necessarily WRONG" mean "sometime is wrong or sometime is right"?
I guess C is wrong, but E sometime is right when f'(1)=-f'(2)=0.

zhouxiaofei170UID: 3185081

I think "necessarily wrong" means "always wrong".
If f is strictly increasing, then for any x1<x2, f(x1)<f(x2).
So f'(x) should always be positive, right?

zhouxiaofei170UID: 3185081

I think I know what you mean now.
A strictly increasing function sometimes can have points whose derivatives are zero.
i.e.     x^3
          cos(x)+x

3楼真是谢谢啦！
：）

zzb198484UID: 3286768

welcome：）

jihadistUID: 3298898

着网站没有Latex我没法看！

spwslfUID: 3216219

难度题！高手来了。

u4475810@anuUID: 3230083

E 是错的 因为strictly increasing 表示导数永远大于0 但是E 不满足
C 是可以的  因为 如果f 在0 到2  都没有定义得话  则 integral  都为0 所以c 存在可能

1990baiyangUID: 2818704

9th floor is right about option C~~ But the explanation of option c I guess should be like:
  "f(x) is strictly increasing" does not mean the integral of f(x) as a function also has to be a strictly increasing function. And in fact, it can be even a decreasing on the domain. The point here is that although f(x) is increasing, it does not have to be positive on the whole space. Hence, if some qualitative change happen in between 0<x<2 like f(x) starts from a negative value and eventually reaches f(x)=0 and then becomes positive. The integral of f(x) can be the same value at some points in the left hand side of f(x)=0 and the right hand side of f(x)=0.
  Example here: Set f(x)=x^2-b (x>=0) let b>0 be any number greater or equal to 2 in this example, then:
  We all know that f(x) is strictly increasing here but the value of f(x) is less than 0 when x<square root of b, hence the integral of f(x) is decreasing. And this quality changes after x becomes greater than square root of b. Therefore...

annimalUID: 2616512

我觉得C 错的。 可以用积分的定义证明呀

E的话 严格递增只需要 导数等于0 的点 不连续 就可以 比如y=x^3 这样的函数在0点邻域确实是严格递增的

treemantanUID: 2770681

Something about inflection point:

treemantanUID: 2770681

C can be proved by Intermediate value of Integration Theorem, Integral of f(x) from 1 to 2 can be equal to 1*f(c)  1<c<2 and in the same way, Integral of f(x) from 0 to 1 is equal to 1*f(d),0<d<1,Since f(c)>f(d) can always be true, the statement is wrong