Question

If the function $f\left(x\right)=3x^4-8x^3+12x^2-48x+25$ has global maximum value $M$ and global minimum value $m$ in $[0,3],$ then $2M+m$ is equal to

Answer 1

$f\left(x\right)=3x^4-8x^3+12x^2-48x+25$
$\Rightarrow f^{\prime }\left(x\right)=12x^3-24x^2+24x-48=12(x^3-2x^2+2x-4)=12(x-2)(x^2+2)$
$f^{\prime }\left(x\right)=0$
$\Rightarrow x=2\in [0,3]$
$f\left(0\right)=25\to M$
$f\left(2\right)=-39\to m$
$f\left(3\right)=16$
So, $2M+m=50-39=11$