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

Let $f:[0,1]\to [0,1]$be a continuous function such that $f\left(f\left(x\right)\right)=1f\text{or}allx\in [0,1]$then:

Let $f:[-\frac{\pi }{2},\frac{\pi }{2}]\to R$ be a continuous function such that $f\left(0\right)=1\text{and}{\int }_0^{\frac{\pi }{3}}f\left(t\right)dt=0$
Then which of the following statement is(are) TRUE?

Let f be a real-valued function defined on interval $(0,\infty )$,by $f\left(x\right)=\ln x+{\int }_0^x\sqrt{1+\sin t}.dt$. Then which of the following statement(s) is (are) true? (A). f"(x) exists for all $\in$ $(0,\infty )$. (B). f'(x) exists for all x $\in$ $(0,\infty )$ and f' is continuous on $(0,\infty )$, but not differentiable on $(0,\infty )$. (C). there exists $\alpha >1$ such that $|f\text{'}\left(x\right)|<\left|f\left(x\right)\right|$ for all x $\in$ $(\alpha ,\infty )$. (D). there exists $\beta >1$ such that $\left|f\left(x\right)\right|+|f\text{'}\left(x\right)|\le \beta$ for all x $\in$ $(0,\infty )$.

Let $f:[0,\infty )\overrightarrow{R}$ be a continuous strictly increasing function, such that $f^3\left(x\right)={\int }_0^{x}t{\stackrel{.}{f}}^2\left(t\right)dt$ for every $x\ge 0.$ Then value of $f\left(6\right)$ is_______

Let $f:[0,1]\to R$ be a continuous function then the maximum value of ${\int }_0^{1}f\left(x\right).x^{2}dx-{\int }_0^{1}x.{\left(f\left(x\right)\right)}^{2}dx$ for all such function(s) is:

Let $f:[0,1]\to R$ be a continuous function then the maximum value of ${\int }_0^{1}f\left(x\right).x^{2}dx-{\in }_0^{1}x.{\left(f\left(x\right)\right)}^{2}dx$ for all such function(s) is:

Let f be a continuous & even function such that $f_0^{a}f\left(x\right)dx=10.\text{if}g\left(x\right)$ is a continuous positive function such that $g\left(x\right)g(-x)=1\text{and}{\int }_0^{a}g\left(x\right)dx=5$ then find the value of ${\int }_{-a}^a\frac{f\left(x\right)}{1+g\left(x\right)}dx$.

Let f be a real-valued function defined on interval $(0,\infty )$,by $f\left(x\right)=\ln x+{\int }_0^x\sqrt{1+\sin t}.dt$. Then which of the following statement(s) is (are) true? (A). f"(x) exists for all $\in$ $(0,\infty )$. (B). f'(x) exists for all x $\in$ $(0,\infty )$ and f' is continuous on $(0,\infty )$, but not differentiable on $(0,\infty )$. (C). there exists $\alpha >1$ such that $|f\text{'}\left(x\right)|<\left|f\left(x\right)\right|$ for all x $\in$ $(\alpha ,\infty )$. (D). there exists $\beta >1$ such that $\left|f\left(x\right)\right|+|f\text{'}\left(x\right)|\le \beta$ for all x $\in$ $(0,\infty )$.