Application Of Derivatives Ques 29

29. Let $f(x)=\int e^{x}(x-1)(x-2) d x$. Then, $f$ decreases in the interval

(a) $(-\infty,-2)$

(b) $(-2,-1)$

(c) $(1,2)$

(d) $(2, \infty)$

(2000, 2M)

Show Answer

Answer:

(b)

Solution:

Formula:

Increasing and decreasing of a function:

  1. Let

$ \begin{aligned} f(x) & =\int e^{x}(x-1)(x-2) d x \\ f^{\prime}(x) & =e^{x}(x-1)(x-2) \\ & +\quad-\quad+ \\ \hline & +\quad 2 \end{aligned} $

$ \Rightarrow \quad f^{\prime}(x)=e^{x}(x-1)(x-2) $

$\therefore f^{\prime}(x)<0$ for $1<x<2$

$\Rightarrow f(x)$ is decreasing for $x \in(1,2)$.

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