Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x3 - 9x2 + 12x + 5 in the interval [0, 3]. Then M - m is equal to :

Application of Derivatives PYQ — JEE Main 2018 · 16 Apr · Shift 1 · Q30 with Solution

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Correct answer: B

Given function
$\quad x\in[0,3]$$ We need the **absolute maximum** $M$ and **absolute minimum** $m$ on $[0,3]$, then compute $M-m$.$
Find critical points inside the interval

Differentiate:

$f'(x)=6x^2-18x+12=6(x^2-3x+2)=6(x-1)(x-2)$

So the critical points are:

$x=1,\; x=2$

Both lie in $[0,3]$ .
Evaluate $f(x)$ at endpoints and critical points

We check:
- $x=0$
- $x=1$
- $x=2$
- $x=3$
$f(0)=2(0)^3-9(0)^2+12(0)+5=5$

$f(1)=2(1)^3-9(1)^2+12(1)+5=2-9+12+5=10$

$f(2)=2(2)^3-9(2)^2+12(2)+5=16-36+24+5=9$

$f(3)=2(3)^3-9(3)^2+12(3)+5=54-81+36+5=14$
Identify absolute maximum and minimum

From the values:

$f(0)=5,\quad f(1)=10,\quad f(2)=9,\quad f(3)=14$
- Absolute maximum value: $M=14$
- Absolute minimum value: $m=5$
Compute $M-m$

$M-m=14-5=9$
Compare with options

$9$ corresponds to Option B.
Verification with stored answer

Stored correct answer: B

Our derived answer also gives B. So the stored answer is correct.