Question

Let the intervals in which the polynomial $2x^3-15x^2+36x+1$ is decreasing be [k,m].find k+m ?

Answer 1

Let $f\left(x\right)=2x^3-15x^2-15x^2+36x+1,$
Then $f^{\prime }\left(x\right)=6x^2-30x+36=6(x-2)(x-3)$ $f^{\prime }\left(x\right)=+iveifx<2orx>3$
$\therefore xϵ[-\infty ,2]orxϵ[3,\infty ]$
Hence $f$ is an increasing function when $xϵ[-\infty ,2]\cup [3,\infty ].$ $f^{\prime }\left(x\right)is-iveif2<x<3$
Hence $f$ is a decreasing function when $xϵ[2,3].$