Question

What are the points of inflection, if any, of #f(x)=x^(1/3)*(x+3)^(2/3) #?

Answer 1

`f(x)=(x)^(1/3)(x+3)^(2/3)`

`f'(x)=(x)^(1/3)(2/3)(x+3)^(-1/3) + (1/3)(x)^(-2/3)(x+3)^(2/3)`
`f'(x)=(2x^(1/3))/(3(x+3)^(1/3)) + (x+3)^(2/3)/(3x^(2/3))`
`f'(x)=(6x+3(x+3))/(9(x+3))`
`f'(x)=(9(x+1))/(9(x+3))`
`f'(x)=(x+1)/(x+3)`

`f''(x)=[(x+3)-(x+1)]/[(x+3)^2]`
`f''(x)=2/(x+3)^2`

No points of inflection of f(x) because you cannot set numerator of the second derivative of f(x) equal to zero.