For what values of x is #f(x)=(2x^2−4x)e^x# concave or convex?

1 Answer
Jan 22, 2018

The function is convex for #x in (-oo,-1-sqrt3) uu(-1+sqrt3, +oo)# and concave for #x in (-1-sqrt3, -1+sqrt3)#

Explanation:

#"Reminder"#

#(uv)'=u'v+uv'#

Calculate the first and second derivatives

#f(x)=(2x^2-4x)e^x=2(x^2-2x)e^x#

#f'(x)=2*(2x-2)e^x+2(x^2-2x)e^x=2(x^2-2)e^x#

#f''(x)=2*(2x)e^x+2(x^2-2)e^x=2(x^2+2x-2)e^x#

The inflection points are when #f''(x)=0#

#x^2+2x-2=0#

Solving this quadratic equation for #x#

#x=(-2+-sqrt(2^2-4*1*(-2)))/(2)=(-2+-sqrt(12))/(2)#

#=(-2+-2sqrt(3))/2#

#=-1+-sqrt3#

The roots are

#x_1=-1-sqrt3#

#x_2=-1+sqrt3#

We can build the variation chart

#color(white)(aaa)##color(white)(aaa)##"Interval"##color(white)(aaa)##(-oo, x_1)##color(white)(aaa)##(x_1, x_2)##color(white)(aaa)##(x_2, +oo)#

#color(white)(aaa)##color(white)(aaa)##"Sign f''(x)"##color(white)(aaaaa)##+##color(white)(aaaaaaaa)##-##color(white)(aaaaaaa)##+#

#color(white)(aaa)##color(white)(aaa)##" f(x)"##color(white)(aaaaaaaaa)##uu##color(white)(aaaaaaaa)##nn##color(white)(aaaaaaa)##uu#

graph{(2x^2-4x)e^x [-17.35, 14.68, -7.95, 8.07]}