To find concavity, use the second derivative of the function.
If #f''(a)<0#, the function is concave down (also called concave) when #x=a#.
If #f''(a)>0#, the function is concave up (also called convex) when #x=a#.
If #f''(a)=0#, the point when #x=a# is when the concavity of a function shifts called the point of inflection.
With your function:
#f'(x)=7/3x^(4/3)+4/3x^(1/3)#
#f''(x)=28/9x^(1/3)+4/9x^(-2/3)#
#f''(x)=(28x+4)/(9x^(2/3))#
graph{(28x+4)/(9x^(2/3)) [-20.28, 20.27, -10.14, 10.14]}
This is a graph of #f''(x)#.
Notice how the #f''(x)=0# when #x=-1/7#. This means that when #x=-1/7# there is a point of inflection on #f(x)#.
#f''(x)<0# when #x<-1/7#, so #f(x)# is concave down whenever #x<-1/7#.
Similarly, #f(x)# is concave up whenever #x> -1/7#.
