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Q.
The total number of parallel tangents of $f_{1}(x)=x^{2}-x+1$ and $f_{2}(x)=x^{3}-x^{2}-2 x+1$ is
Application of Derivatives
Solution:
Here, $f_{1}(x)=x^{2}-x+1$ and $f_{2}(x)=x^{3}-x^{2}-2 x+1$
or $f_{1}'\left(x_{1}\right)=2 x_{1}-1$ and $f_{2}'\left(x_{2}\right)=3 x_{2}{ }^{2}-2 x_{2}-2$
Let the tangents drawn to the curves $y=f_{1}(x)$ and $y=f_{2}(x)$
at $\left(x_{1}, f_{1}\left(x_{1}\right)\right)$ and $\left(x_{2}, f_{2}\left(x_{2}\right)\right)$ be parallel. Then
$2 x_{1}-1=3 x_{2}^{2}-2 x_{2}-2$
or $2 x_{1}=\left(3 x_{2}^{2}-2 x_{2}-1\right)$
which is possible for infinite numbers of ordered pairs.
So, there are infinite solutions.