UPSC NDA Math Model Paper 8

Given: f(x)=(x+2)e−x
Here we have to find the interval in which f(x) is decreasing.
Let's first calculate f′(x)
As we know that,

d

dx

[f(x).g(x)]=f(x).

d{g(x)}

dx

+g(x).

d{f(x)}

dx

⇒f′(x)=(x+2).

d(e−x)

dx

+e−x.

d(x+2)

dx

As we know that,

d(eax)

dx

=a.eax
⇒f′(x)=−(x+2).e−x+e−x.1
⇒f′(x)=−(x+1).e−x
As we know that for a decreasing function say f(x) we have f′(x)≤0
⇒−(x+1).e−x≤0
⇒(x+1).e−x≥0
⇒x+1≥0 ........(∵e−x>0)
⇒x≥−1
Hence, the interval in which the given function is decreasing is [−1,∞)