Question

Let $f$ be a twice differentiable function defined on $R$ such that $f (0)=1, f'(0)=2$ and $f'(x)\neq 0$ for all $x \in R$. If $\left|\begin{array}{cc}f(x) & f'(x) \\ f'(x) & f''(x)\end{array}\right|=0,$ for all $x \in R,$ then the value of $f (1)$ lies in the interval:

• A. (9,12)
• B. (6,9)
• C. (0,3)
• D. (3,6)

Answer 1

Answer: (B)

$f(x) f''(x)-\left(f'(x)\right)^{2}=0$
$\frac{f''(x)}{f'(x)}=\frac{f'(x)}{f(x)}$
ln $\left(f'(x)\right)=\ln f(x)+\ln c$
$f'(x)= cf (x)$
$\frac{f'(x)}{f(x)}=c$
lnf$(x)=c x+k_{1}$
$f(x)=k e^{c x}$
$f(0)=1=k$
$f'(0)=c=2$
$f(x)=e^{2 x}$
$f(1)=e^{2} \in(6,9)$