Question

Let $f:[0,10]\to [0,\infty )$ be a continuous function such that ${\int }_0^{1}f\left(x\right)dx=10$. Which of the following statements is NOT necessarily true?

• A. ${\int }_0^1{e}^{-x}f\left(x\right)dx\le 10$
• B. ${\int }_0^1\frac{f\left(x\right)}{(1+x{)}^2}dx\le 10$
• C. $-10\le {\int }_0^1\sin \left(100x\right)f\left(x\right)dx\le 10$
• D. ${\int }_0^{1}f(x{)}^{2}dx\le 100$

Answer 1

The correct option is D ${\int }_0^{1}f(x{)}^{2}dx\le 100$

$f:[0,1]\to [0,\infty )\Rightarrow f\left(x\right)\ge 0$

${\int }_0^{1}f{\left(x\right)}^{2}dx\le 100$ is not necessarily true

Because ${\left(f\left(x\right)\right)}^2$ can take very high values. Then, the area bounded by ${\left(f\left(x\right)\right)}^2$ and $x$ axis and from $x=0$ to $x=1$ may cross $100$.

In all the other options there is a function muliplied with $f\left(x\right)$ that may compensate the whole value of the function.

Hence, option $D$ is correct