Differential Calculus
Intervals of increase/decrease via derivative
MJMT_Full_Test_01
Grade 12
Question:
Let $f(x)=\displaystyle\int e^x(x-1)(x-2)\,dx$. Then $f(x)$ decreases in the interval
$(2,\infty)$
$(-2,-1)$
$(1,2)$
$(-\infty,1)\cup(2,\infty)$
Step-by-Step Solution
Key Concept: $f'(x)=e^x(x-1)(x-2)$. Since $e^x>0$, sign of $f'$ equals sign of $(x-1)(x-2)$, which is negative on $(1,2)$.
$f'(x)<0$ iff $(x-1)(x-2)<0$ iff $x\in(1,2)$.
Correct Answer: 3