Why Option B) is incorrect?

Asked Feb 09 '20 at 18:04

Active Feb 09 '20 at 18:48

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Let $f\left(x\right)\le x^5$ be a continuous function for all x $\epsilon$ R then $f\left(x\right)$ is

A) All answers are wrong

B) The integral average of $f(x)$ is on $[0,1]$ is positive

C) The integral average of $f(x)$ is on $[0,1]$ is negative

D) The integral average of $f(x)$ is on $[-1,0]$ is positive

E) The integral average of $f(x)$ is on $[-1,0]$ is negative

for my opinion , Both B) and E) are right since I applied integral formula on $x^5$ giving two results

edited Feb 09 '20 at 18:48

asked Feb 09 '20 at 18:04

Feemo Fellow

Technically, A) runs into the liar paradox :) – Maximilian Janisch Feb 09 '20 at 18:12
Ignoring that, E) is the only right answer ( C) is wrong, consider $f(x)=x^5$ and D) is wrong, consider also $f(x)=x^5$) – Maximilian Janisch Feb 09 '20 at 18:14
E is the correct answer, because $f$ is bounded above by function whos integral is negative on $[-1,0]$ – fGDu94 Feb 09 '20 at 18:16
I don't understand I meant to write C and E but it was confusion which to select – Feemo Fellow Feb 09 '20 at 18:20
@fGDu94 but if we take [0,1] why we cannot say that is bounded above by function whos integral is on [0,1] is positive – Feemo Fellow Feb 09 '20 at 18:31
sure we can, its just that $f$ could be for example $f(x) = -1$ on $[0,1]$, so we can't be sure has a positive integral on $[0,1]$. Meanwhile you could have $f(x) = x^5$ on $[0,1]$ as well, which is positive. – fGDu94 Feb 09 '20 at 18:33
can I go for option c also then – Feemo Fellow Feb 09 '20 at 18:34
no because $f(x) = x^5$ is a counterexample – fGDu94 Feb 09 '20 at 18:35
I now understand sir but last question f(x)=1 on [-1,0] we also cannot be sure – Feemo Fellow Feb 09 '20 at 18:42
$f(x) \leq x^5$ on $[-1,0]$.
Therefore $\int_{-1}^0 f(x) \leq \int_{-1}^0 x^5 dx = \frac{-1}{6} < 0$.
– fGDu94 Feb 09 '20 at 18:53
thanks sir got it , b is incorrect because f(x) can benegative on [0,1] and e is right because f(x) is negative on [-1,0] so we can be sure – Feemo Fellow Feb 09 '20 at 19:03

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