Question

Let $M$ and $m$ be respectively the absolute maximum and the absolute minimum values of the function, $f\left(x\right)=2x^3-9x^2+12x+5$ in the interval $[0,3]$. Then $M-m$ is equal to.

• A. $1$
• B. $5$
• C. $4$
• D. $9$

Answer 1

The correct option is D $9$

$f\left(x\right)=2x^3-9x^2+12x+5$

$f^{{}^{\prime }}\left(x\right)=6x^2-18x+12=0$

$⟹x^2-3x+2=0$

$x=1$ or $x=2$

Thus, $f\left(1\right)=2-9+12+5=10$

And $f\left(2\right)=2(2{)}^3-9(2{)}^2+12\left(2\right)+5=16-36+24+5$

Thus $f\left(2\right)=9$

Now, also we need to check the values of the function at $x=0$ and $x=3$

$f\left(0\right)=5$

Also, $f\left(3\right)=2(3{)}^3-9(3{)}^2+12\left(3\right)+5$

Thus $f\left(3\right)=95-81=14$

$f^{{}^{\prime \prime }}\left(x\right)=12x-18$

Thus $M=14$ and $m=5$ since $M$ and $m$ are the absolute Maxima and minima of the given function.

Thus $M-m=14-5=9$