Question

If a function $y=f\left(x\right)$ is such that $f^{\prime }\left(x\right)$ is continuous function and satisfies $\left(f\right(x){)}^2=K+{\int }_0^x\left(\right(f\left(t\right){)}^2+\left(f^{\prime }\right(t\left){)}^2\right)dt,KϵR^{+}$, then

• A. f(x) is increasing function $\forall xϵR$
• B. f(x) is bounded function
• C. f(x) is neither even nor odd
• D. if $K=100$ then $f\left(0\right)=10$

Answer 1

Answer: (A), (C), (D)

The correct options are
A

f(x) is increasing function $\forall xϵR$

C

f(x) is neither even nor odd

D

if $K=100$ then $f\left(0\right)=10$

Differentiating the given equation $2f\left(x\right)f^{\prime }\left(x\right)=\left(f\right(x){)}^2$ + $\left(f^{\prime }\right(x){)}^2\Rightarrow$ (f(x) – $f^{\prime }\left(x\right){)}^2=0$

$\Rightarrow f^{\prime }\left(x\right)=f\left(x\right)\Rightarrow f\left(x\right)=c{e}^x$

$\because f\left(0\right)=\sqrt{K}\Rightarrow c=\sqrt{K}\Rightarrow f\left(x\right)=\sqrt{K}{e}^x$