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 the value of $M-m$ is equal to:

• A. $5$
• B. $9$
• C. $10$
• D. $1$

Answer 1

The correct option is B $9$

Given:

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

in the interval [0,3]

Solution:

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

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

Putting f'(x) = 0, we get

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

$\Rightarrow 6(x^2-3x+2)=0$

$\Rightarrow (x-1)(x-2)=0$

$\Rightarrow x=1,2$

Thus,

$f\left(1\right)=2(1{)}^3-9(1{)}^2+12\left(1\right)+5=10$

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

We also need to find the values of f(x) at x=0 and x=3,

$f\left(0\right)=2(0{)}^3-9(0{)}^2+12\left(0\right)+5=5$

$f\left(3\right)=2(3{)}^3-9(3{)}^2+12\left(3\right)+5=14$

Thus M=14 and m=5.

$\therefore M-m=14-5=9$