Question

Let $f\left(x\right)=\int e^x(\mathrm{x}-1)(\mathrm{x}-2)\mathrm{dx}$ Then$\mathrm{f}$decreases in the interval

• A. $(-\infty ,-2)$
• B. $(-2,-1)$
• C. $(1,2)$
• D. $(2,\infty )$

Answer 1

The correct option is C

$(1,2)$

Explanation for the correct option:

Given that: $\mathrm{f}\left(x\right)=\int {\mathrm{e}}^{\mathrm{x}}·(\mathrm{x}-1)(\mathrm{x}-2)\mathrm{dx}.$

For a decreasing function, ${\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)<0$

Lets take derivative of given function, we have
$$\begin{gathered} \therefore f^{\text{'}}\left(x\right)=\frac{d}{dx}\int e^x(x-1)(x-2)dx \\ {f}^{\text{'}}\left(x\right)=e^x(x-1)(x-2) \end{gathered}$$

Now,

$$\begin{gathered} e^x(x-1)(x-2)<0 \\ \Rightarrow (x-1)(x-2)<0 \\ \Rightarrow x\in (1,2) \end{gathered}$$ Because exponential function can never less than zero so, the decreasing interval is $\left(1,2\right)$

Hence, the correct answer is option (C).