Question

Let $f\left(x\right)=3x^2-7x+c$ where ${}^{\prime }{c}^{\prime }$ is a parameter and $x\ge \frac{7}{6}$. Then the value of $\left[c\right]$ such that $f\left(x\right)$ touches $f^{-1}\left(x\right)$ is :
([.] represents the greatest integer function)

Answer 1

$f\left(x\right)$ and $f^{-1}\left(x\right)$ can only intersect on the line $y=x$ and hence $y=x$ must be tangent at the common point of tangency.
$\therefore 3x^2-7x+c=x$
$\Rightarrow 3x^2-8x+c=0$
This equation must have equal roots.
$\Rightarrow 64-12c=0$
$\Rightarrow c=\frac{64}{12}=\frac{16}{3}$
$\therefore \left[c\right]=5$