Let M and m be respectively the local maximum and the local minim… - Vidyadip

MCQ

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

✓
$1$
B
$5$
C
$4$
D
$9$

✓

Answer

Correct option: A.

$1$

a
Here, $f(x)=2 x^{3}-9 x^{2}+12 x+5$

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

For maxima or minima put $f^{\prime}(x)=0$

$\Rightarrow x^{2}-3 x+2=0$

$\Rightarrow x=1$ or $x=2$

Now, $f^{\prime \prime}(x)=12 x-18$

$\Rightarrow f^{\prime \prime}(1)=12(1)-18=-6<0$

Hence, $f(x)$ has maxima at $x=1$

$\therefore$ maximum value

$=M=f(1)=2-9+12+5$ $=10$.

And, $f^{\prime \prime}(2)=12(2)-18=6>0$.

Hence, $f(x)$ has minima at $x=2$

$\therefore$ minimum value $=m=f(2)=2(8)-9(4)+$

$12 (2)$

$+5=9$

$\therefore M-m=10-9=1$

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