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For what values of x is #f(x)=(x-2)(x-4)(x-3)# concave or convex?

1 Answer

Nov 22, 2016

#f(x)# is concave on (-oo,3), and convex on #(3,oo)#.

Explanation:

To find where a function is concave or convex, find where #f''(x)# is positive or negative (respectively).

#f(x)=(x-2)(x-4)(x-3)#

#f'(x)=(1)(x-4)(x-3)+(x-2)(1)(x-3)+(x-2)(x-4)(1)#
#f'(x)=x^2-7x-12+x^2-5x+6+x^2-6x+8#
#f'(x)=3x^2-18x+2#

#f''(x)=6x-18#
#f''(x)=6(x-3)#

#f''(x)# is negative from #(-oo,3)#, and positive from #(3,oo)#

Therefore, #f(x)# is concave on (-oo,3), and convex on #(3,oo)#.

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