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For what values of x is #f(x)=(2x-1)(3x-5) (x-2)# concave or convex?

1 Answer
Mar 8, 2017

Answer:

#f(x)# is concave when #x in ]-oo,25/18[# and convex when #x in ]25/18,+oo[#

Explanation:

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[#