For what values of x is #f(x)=x^3e^x# concave or convex?

Question

Emily Ammerman · Accepted Answer

The function is concave for #x in (-oo,-3-sqrt3)uu(-3+sqrt3,0)# and convex for #x in (-3-sqrt3, -3+sqrt3)uu(0,+oo)#

Calculate the first and second derivative by the product rule

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

#f(x)=x^3e^x#

#f'(x)=3x^2e^x+x^3e^x=(x^3+3x^2)e^x#

#f''(x)=(3x^2+6x)e^x+(x^3+3x^2)e^x#

#=x(x^2+6x+6)e^x#

The points of inflections are when, #f''(x)=0#

#x(x^2+6x+6)e^x#, #e^x>0#

#=>#, #x(x^2+6x+6)=0#

#=>#, #{(x=0),(x^2+6x+6=0):}#

#=>#, #x=(-6+-sqrt(36-24))/(2)=-3+-sqrt3#

There are #3# points of inflections

Therefore,

There are #4# intervals to consider are

#I_1=(-oo,-3-sqrt3)# and #I_2=(-3-sqrt3, -3+sqrt3)# and #I_3=(-3+sqrt3,0)# and #I_4=(0,+oo)#

Let's consider a variation chart

#color(white)(aaaa)##"Interval"##color(white)(aaaaaa)##I_1##color(white)(aaaaa)##I_2##color(white)(aaaa)##I_3##color(white)(aaaa)##I_4#

#color(white)(aaaa)##"sign f''(x)"##color(white)(aaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaa)##+#

#color(white)(aaaa)##" f(x)"##color(white)(aaaaaaaa)##nn##color(white)(aaaa)##uu##color(white)(aaaa)##nn##color(white)(aaaa)##uu#

The function is concave for #x in (-oo,-3-sqrt3)uu(-3+sqrt3,0)# and convex for #x in (-3-sqrt3, -3+sqrt3)uu(0,+oo)#

graph{x^3e^x [-10, 10, -5, 5]}

Christopher Agresta · Answer

undefined To determine the concavity or convexity of the function $ f(x) = x^3e^x $, you need to find its second derivative, $ f''(x) $, and then analyze the sign of $ f''(x) $ to identify the intervals where the function is concave up (convex) or concave down.

1. Find the first derivative of $ f(x) $:
   $ f'(x) = (3x^2)e^x + (x^3)e^x $

2. Find the second derivative $ f''(x) $ by differentiating $ f'(x) $:
   $ f''(x) = (6x)e^x + (3x^2)e^x + (3x^2)e^x + (x^3)e^x $

3. Simplify $ f''(x) $:
   $ f''(x) = (x^3 + 6x + 6x^2)e^x $

4. To determine the concavity or convexity of $ f(x) $, analyze the sign of $ f''(x) $:
   - If $ f''(x) > 0 $, the function is concave up (convex) on that interval.
   - If $ f''(x) < 0 $, the function is concave down on that interval.

5. The expression $ (x^3 + 6x + 6x^2)e^x $ is always positive for all real values of $ x $. Therefore, $ f(x) = x^3e^x $ is always concave up (convex) for all values of $ x $.