Question

The total number of parallel tangents of $f_1(x)=x^2-x+1$ and $f_2(x)=x^3-x^2-2x+1$ is

• A. 2
• B. 0
• C. 1
• D. infinite

Answer 1

Answer: (D)

Here, $f_1(x)=x^2-x+1$ and $f_2(x)=x^3-x^2-2x+1$
or $f_1^{\prime }\left(x_1\right)=2x_1-1$ and $f_2^{\prime }\left(x_2\right)=3x_2{}^2-2x_2-2$
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
$2x_1-1=3x_2^2-2x_2-2$
or $2x_1=\left(3x_2^2-2x_2-1\right)$
which is possible for infinite numbers of ordered pairs.
So, there are infinite solutions.