If a function $y=f(x)$ such that $f'(x)$ is continuous function and satisfies

$(f(x))^2=k+\int\limits_0^x\left[\{f(t)\}^2+\left\{f'(t)\right\}^2\right] d t, k \in R^{+}$, then

(a) $f(x)$ is an increasing function for all $x \in R$
(b) $f(x)$ is a bounded function
(c) $f(x)$ is neither even nor odd function
(d) If $k=100$, then $f(0)=10$.

(a), (c), (d)

We have

$(f(x))^2=k+\int\limits_0^x\left[f^2(t)+f'^2(t)\right] dt$           ......(1)

Differentiating with respect to $x$

$2 f(x) f'(x)=f^2(x)+f'^2(x)$

so $\left(f'(x)-f(x)\right)^2=0$

so $f'(x)=f(x)⇒f'(x)/f(x)=1$

so $\log f(x)=x+\log c$   ....(2)

at $x = 0$ from (1)

$f^2(0)=k⇒f(0)=\sqrt{k}$

from (2) $\log f(0)=0=0+\log c$

so $\log \sqrt{k}=\log c⇒c=\sqrt{k}$

from (2) $\log f(x)=x+\log c$

so $f(x)=\sqrt{k}e^x$

f(x) is increasing for all $x∈R$ and not Bounded and neither odd/even

$f(0) = \sqrt{100}e^0=10$  (at k = 100)