MCQ
Suppose $\left| {\begin{array}{*{20}{c}}
  {f'\left( x \right)}&{f\left( x \right)} \\ 
  {f''\left( x \right)}&{f'\left( x \right)} 
\end{array}} \right| = 0$ where $f(x)$ is continuously differentiable function with $f'(x) \ne  0$ and satisfy $f(0) = 1$ and $f'(0) = 2$ , then the number of solution $(s)$ of equation $f(x) = x^2$ is equal to 
  • A
    $0$
  • $1$
  • C
    $2$
  • D
    $3$

Answer

Correct option: B.
$1$
b
$\left(f^{\prime}(\mathrm{x})\right)^{2}-f(\mathrm{x}) f^{\prime \prime}(\mathrm{x})=0 \Rightarrow \frac{\mathrm{d}}{\mathrm{d} \mathrm{x}}\left(\frac{f(\mathrm{x})}{f^{\prime}(\mathrm{x})}\right)=0$

$ \Rightarrow \frac{{f({\rm{x}})}}{{{f^\prime }({\rm{x}})}} = $ constant

$\Rightarrow \frac{f(\mathrm{x})}{f^{\prime}(\mathrm{x})}=\frac{1}{2} \Rightarrow f(\mathrm{x})=\mathrm{e}^{2 \mathrm{x}}$

The equation $e^{2 x}=x^{2}$ has one solution.

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