We develop the expression and we calculate the first and second derivatives
#f(x)=(2x-1)(3x-5)(x-2)#
#=(6x^2-13x+5)(x-2)#
#=6x^3-12x^2-13x^2+26x+5x-10#
#f(x)=6x^3-25x^2+31x-10#
#f'(x)=18x^2-50x+31#
#f''(x)=36x-50#
#f''(x)=0# when #x=50/36=25/18#
We construct a table
#color(white)(aaaa)##Interval##color(white)(aaaa)##|##color(white)(aaaa)##]-oo,25/18[##color(white)(aaaa)##|##color(white)(aaa)##]25/18,+oo[##|#
#color(white)(aaaa)##Sign f''(x)##color(white)(aa)##|##color(white)(aaaaaaaa)##-##color(white)(aaaaaaa)##|##color(white)(aaaaaaaa)##+##color(white)(aa)##|#
#color(white)(aaaa)##function##color(white)(aaaa)##|##color(white)(aaaaaaaaa)##nnn##color(white)(aaaaaa)##|##color(white)(aaaaaaaa)##uuu##color(white)(aa)##|#
Therefore,
#f(x)# is concave when #x in ]-oo,25/18[# and convex when #x in ]25/18,+oo[#