The curves `y=4x^(2)+2x-8` and `y=x^(3)-x+13` touch each other at the point

Explanation

The curves `y=4x^(2)+2x-8` and `y=x^(3)-x+13` touch each other at the pont `(3,34)`
Give, equation of curves are `y=4x^(2)+2x-8` and `y=x^(3)-x+13`
`therefore (dy)/(dx)=8x+2` ltbgt and `(dy)/(dx)=3x^(2)-1`
So, the slope of both curves should be same
`therefore 3x^(2)-9x+x-3=0`
`rArr 3x(x-3)+1(x-3)=0`
`rArr (3x+1)(x-3)=0`
`therefore x=-1/3` and x=3.
For x`=-1/3`, `y=4.(-1/3)^(2)+2.(-1/3)-8`
`=4/9-2/3-8=(4-6-72)/(9)`
`=-74/9`
and for x=3, y=`4.(3)^(2)+2.(3)-8= 36+6-8=34`
So, the required points are `(3,34)` and `(-1/3, -79/9)`.