Application of Derivatives 2 Question 12
12. 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
Solution:
- 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 $x \in(1,2)$.
