Question

f(x)= x sqrt(x + 4). . which intervals is it increasing and decreasing on?. which interval is it concave up?

Answer 1

a ) The domain : x ∈ ( - 4 , + ∞ )
f ` ( X ) =
`√(x+4)+ x * (1)/(2 √(x+4) )= \\ (2x+8+x)/(2 √(x+4) )= \\ (3x+8)/(2 √(x+4) )`
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 ) =
`(1)/(2) * (3 √(x+4)- (3x+8)/(2 √(x+4) ) )/(x+4) \\ = (1)/(4)* (6x+24-3x-8)/( (x+4)^(3/2) ) = (1)/(4) * (3x+16)/((x+4) ^(3/2) )`
f `` ( x ) > 0:
3 x + 16 > 0
x > - 16/3
x > - 5 1/3
f ( x ) is concave up for x ∈ ( - 4 , + ∞ ) - the domain