Suppose | * 20 c f' ( x ) & f ( x ) f'' ( x ) & f' ( x ) | = 0 wh… - Vidyadip

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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