Find points of inflection and discuss concavity for f(x)=x(x-4)3+k. Use k=2 example provided by teacher: SOLUTION: Lets use the product and chain rule to find f(x)= x*3(x-4)^2 + (x-4)3; factor out the common binomial to get f(x)=[3x+x-4](x-4)2= [4x-4](x-4)2= 4[x-1](x-4)2; Now to f lets use the product and chain rule again. f(x)=4[x-1]*2(x-4) + 4(x-4)2; factor out the 4(x-4) to get f(x)=4(x-4){2x-2+x-4}=4(x-4){3x-6}=12(x-4){x-2}. Setting this equal to zero gives the point of inflection as x=2 and x=4. Lets take f(3) to see the concavity there. SOLUTION: f(3)<0 findings