Question

If the greatest value of the function $f\left(x\right)=\frac{1}{4}{x}^4-\frac{2}{3}{x}^3-\frac{3}{2}{x}^2+2;x\in [-2,4]$ is M, then find the value of [M]; where [ ] denotes greatest integer function.

Answer 1

$f\left(x\right)=\frac{x^4}{4}-\frac{2x^3}{3}-\frac{3x^2}{2}+2$
$\Rightarrow f^{\prime }\left(x\right)=x^3-2x^2-3x$
$=x(x-3)(x+1)$
Maxima at $x=0$
$f\left(0\right)=2$
$\Rightarrow f(-2)=4+\frac{16}{3}-6+2=\frac{16}{3}$

$\Rightarrow f\left(4\right)=64-\frac{128}{3}-24+2=-\frac{2}{3}$

But $f(-2)=\frac{16}{3}$ is the greatest value of $f$

$\Rightarrow \left[\frac{16}{3}\right]=5$ is the greatest value.