Question

If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .

Options

• \[1 \text { for x } < - 3\]

• \[- 1\text {  for x} < - 3\]

• \[1\text {  for all } x \in R\]

• none of these

Answer 1

\[- 1\text {  for x} < - 3 \]

 

\[\text { We have, }f\left( x \right) = \sqrt{x^2 + 6x + 9}\]

\[ = \sqrt{\left( x + 3 \right)^2} \]

\[ = \left| x + 3 \right| \]

`f(x) ={(x+3, x>=-3) ,(-x-3, x<-3):}`

`rArrf'(x)={[1, x>=-3], [-1, x<-3]:}`

`thereforef'(x)=-1` for `x<-3`