We need to use the second derivative test.
f(x)=0∫x(2t3−15t2+36t) dt
f′(x)=2x3−15x2+36x
f′′(x)=6x2−30x+36
Finding points of inflection:
0=6x2−30x+36
0=x2−5x+6
0=(x−3)(x−2)
Testing x=2 and x=3:
Interval Test Value Sign of f′′(x)Conclusion−∞<x<2x=−4f′′(−4)>0Concave up2<x<3x=2.5f′′(2.5)<0Concave down3<x<∞x=4f′(4)>0concave up