Question

Question #db8e2

Answer 1

For increasing you have to find out where the first derivative is positive, for decreasing where first derivative is negative

Explanation:

Let's calculate the first derivative of the function `f(x)`:

`[(x+2) * e^-x]'` = `(x+2)' * e^-x+(x+2)*(e^-x)'=1*e^-x+(x+2) (-e^-x)=e^-x-xe^-x-2e^-x=-xe^-x-e^-x=e^-x (-x-1)`

Now, regardless of the value of `x`, `e^-x` is always positive, so the sign of the first derivative depends on whether `(-x-1)` is positive or negative. That is:

• If `(-x-1)>0`, the first derivative is positive and the function is increasing. Thus, if `-1>x` the function is increasing
• Conversely, if `(-x-1)<0`, the first derivative is negative and the function is decreasing. Thus, if `-1 < x` the function is decreasing