Determine the derivative of #f(x)#
#f' (x)#
#=((x-2)(x-4)^3*1-x[(x-2)*3(x-4)^2 +(x-4)^3 *1] )/[(x-2)(x-4)^3]^2#
Take the numerator then equate to zero
#((x-2)(x-4)^3*1-x[(x-2)*3(x-4)^2 +(x-4)^3 *1] )=0#
simplify
#(x-2)(x-4)^3-3x(x-2)(x-4)^2-x(x-4)^3=0#
Factoring the common term
#(x-4)^2*[(x-2)(x-4)-3x(x-2)-x(x-4)]=0#
#(x-4)^2*(x^2-6x+8-3x^2+6x-x^2+4x)=0#
#(x-4)^2(-3x^2+4x+8)=0#
The values of x are:
#x=4# an asymptote
#x_1=(4+sqrt(112))/6=2.430500874043#
Use #x_1# to obtain #y_1=-1.4602879768904# Maximum
#x_2=(4-sqrt(112))/6=-1.0971675407097#
Use #x_2# to obtain #y_2=-0.002674986072485## Minimum
