Question

Select the correct statement fiven option for the function $f\left(x\right)=x{e}^{1-x}$

• A. Strictly increases in the interval $\left(\frac{1}{2},2\right)$
• B. Increases in the interval $\left(0,\infty \right)$
• C. Decreases in the interval $\left(0,2\right)$
• D. Strictly decreases in the interval $(1,\infty )$

Answer 1

The correct option is D

Strictly decreases in the interval $(1,\infty )$

Finding the derivative of the given function and finding the interval:

$f\left(x\right)=x{e}^{1-x}$

Differentiating the given function

$$\begin{gathered} f\text{'}\left(x\right)=–x{e}^{1-x}+e^{1-x} \\ =e^{1-x}(1-x) \end{gathered}$$

$f\text{'}\left(x\right)<0,\forall (1,\infty )$

$f\left(x\right)$strictly decreases in the interval $(1,\infty )$

Therefore, the correct answer is option (D).