For what values of x is #f(x)= 2x^3+5x+12# concave or convex?

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Answer 1 · 2017-03-07T09:26:35

#f(x)# is concave when #x in ]-oo,0[#
#f(x)# is convex when #x in ]0,+oo[#

We calculate the first and second derivatives

#f(x)=2x^3+5x+12#

#f'(x)=6x^2+5#

#f''(x)=12x#

#f'(x)>0#

#f''(x)=0#, when #x=0#

We draw a chart

#color(white)(aaaa)##Interval##color(white)(aaaaaaa)##]-oo,0[##color(white)(aaaa)##]0,+oo[#

#color(white)(aaaa)##Sign f''(x)##color(white)(aaaaaaaa)##-##color(white)(aaaaaaaa)##+#

#color(white)(aaaa)##function##color(white)(aaaaaaaaaa)##nnn##color(white)(aaaaaaaa)##uuu#

Therefore,

#f(x)# is concave when #x in ]-oo,0[#

#f(x)# is convex when #x in ]0,+oo[#