Question

If f(x) is real valued continuous and differentiable satisfying $\left(f\right(x){)}^2={\int }_0^x\left(f^2\right(t)+(f^{\prime }\left(t\right){)}^2)dt+2013$ then which of following are possible.

• A. $f\left(x\right)=\left(\sqrt{2013}\right)e^x$
• B. $f\left(x\right)=(-\sqrt{2013})e^x$
• C. $f\left(x\right).f^{\prime }\left(x\right)=\left(f^{\prime \prime }\right(x){)}^2$
• D. $f\left(x\right)=(-\sqrt{2013})+2x$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $f\left(x\right)=\left(\sqrt{2013}\right)e^x$
B $f\left(x\right)=(-\sqrt{2013})e^x$
C $f\left(x\right).f^{\prime }\left(x\right)=\left(f^{\prime \prime }\right(x){)}^2$

$2f\left(x\right)f^{\prime }\left(x\right)=f^2\left(x\right)+f^{12}\left(x\right)$
$\left(f^1\right(x)-f(x){)}^2=0$ $f^{\prime }\left(x\right)=f\left(x\right)$
$f\left(x\right)=C{e}^x$