a ) The domain : x ∈ ( - 4 , + ∞ )
f ` ( X ) =[tex] \sqrt{x+4}+ x * \frac{1}{2 \sqrt{x+4} }= \\ \frac{2x+8+x}{2 \sqrt{x+4} }= \\ \frac{3x+8}{2 \sqrt{x+4} } [/tex]
f ` ( x ) < 0
3 x + 8 < 0
x < - 8 /3
x < -2 2/3
f ( x ) is decreasing for : x ∈ ( - 4 , - 2 2/3 )
f ( x ) is increasing for: x ∈ ( -2 2/3, + ∞ )
b ) f `` ( x ) = [tex] \frac{1}{2} * \frac{3 \sqrt{x+4}- \frac{3x+8}{2 \sqrt{x+4} } }{x+4} \\ = \frac{1}{4}* \frac{6x+24-3x-8}{ (x+4)^{3/2} } = \frac{1}{4} * \frac{3x+16}{(x+4) ^{3/2} } [/tex]
f `` ( x ) > 0:
3 x + 16 > 0
x > - 16/3
x > - 5 1/3
f ( x ) is concave up for x ∈ ( - 4 , + ∞ ) - the domain
