Question

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1,f^{\prime }(0)=2$ and $f^{\prime }(x)\ne 0$ for all $x\in R$. If $\left|\begin{array}{cc}f(x)& f^{\prime }(x)\\ f^{\prime }(x)& f^{\prime \prime }(x)\end{array}\right|=0,$ 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)

$f(x)f^{\prime \prime }(x)-{\left(f^{\prime }(x)\right)}^2=0$
$\frac{f^{\prime \prime }(x)}{f^{\prime }(x)}=\frac{f^{\prime }(x)}{f(x)}$
ln $\left(f^{\prime }(x)\right)=\ln f(x)+\ln c$
$f^{\prime }(x)=cf(x)$
$\frac{f^{\prime }(x)}{f(x)}=c$
lnf$(x)=cx+k_1$
$f(x)=k{e}^{cx}$
$f(0)=1=k$
$f^{\prime }(0)=c=2$
$f(x)=e^{2x}$
$f(1)=e^2\in (6,9)$