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For what values of x is #f(x)= -x^4-9x^3+2x+4 # concave or convex?

Answer 1

Emilia Aguilar

Convex for #x in (-9/2 , 0)#

Concave for #x in (-oo , -9/2 )uu(0,oo)#

A function is convex where its second derivative #f''>0#, and concave where its second derivative #f''<0#

The second derivative is the derivative of the first derivative .i.e.

#f''=f'(f')#

#f'(x)=-4x^3-27x^2+2#

#f''(x)=f'(f'(x)=f'(-4x^3-27x^2+2)=-12x^2-54x#

Convex:

#-12x^2-54x>0#

#x(-12x-54)>0#

#x>0#

#-12x-54>0#

#-12x>54#

#x<-54/12#

#x<-9/2#

#x<0#

#-12x-54<0#

#x> 54/-12# , #x> -9/2#

Convex for #x in (-9/2 , 0)#

Concave:

#-12x^2-54x<0#

#x(-12x-54)<0#

#x<0#

#-12x-54<0# , #x> -9/2#

#x>0#

#-12x-54>0# , #x<-9/2#

Concave for #x in (-oo , -9/2 )uu(0,oo)#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.