Question
Let $$f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.} $$ Then $$f$$ decreases in the interval
A.
$$\left( { - \infty , - 2} \right)$$
B.
$$\left( { - 2, - 1} \right)$$
C.
$$\left( {1,2} \right)$$
D.
$$\left( {2, + \infty } \right)$$
Answer :
$$\left( {1,2} \right)$$
Solution :
$$\eqalign{
& f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx} \cr
& {\text{For decreasing function, }}f'(x) < 0 \cr
& \Rightarrow {e^x}(x - 1)(x - 2) < 0 \Rightarrow \,\,\,(x - 1)(x - 2) < 0 \cr
& \Rightarrow 1 < x < 2,\quad \because {e^x} > 0\forall x \in R \cr} $$