Function f ( x ) = _1^x 2 ( t - 1 ) ( t - 2 ) ^3 + 3 ( t - 1 ) ^2… - Vidyadip

MCQ

Function

$f\left( x \right) = \int_1^x {\left\{ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right\}} dt$ is maximum when $x$ is equal to

✓
$1$
B
$2$
C
$3$
D
$4$

✓

Answer

Correct option: A.

$1$

a
$f ( x )=\int_{1}^{ x }\left(2( t -1)( t -2)^{3}+3(t-1) 2( t -2)^{2}\right) d t=\int_{1}^{x} 2(t-1)(t-2)^{3}$

$\Rightarrow f^{\prime}(x)=2(x-1)(x-2)^{3}=0$ for extrema. $\Rightarrow x=1,2$

Also $f^{\prime}(x)=2(x-2)^{2}+6(x-1)(x-2)^{2}$

Clearly $f^{\prime \prime}(1)<0$ and $f^{\prime \prime}(2)=0$

Hence $x=1$ is the point of maxima.

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