Consider the function f ( x )= 27 ( x ^ 2 / 3 - x ) 4 How many solutions does the function f ( x )=1 have?

Step-by-step solution

aligned f(x) & = 27 4 (x^ 2 / 3 -x ) & = 27 4 ( (x^2 )^ 1 / 3 -x ) aligned For f(x)=1 aligned & 27 4 ( (x^2 )^ 1 / 3 -x )=1 & (x^ 1 / 3 )^2-x= 4 27 aligned Put x^ 1 / 3 =yx=y^3 aligned & y^2-y^3= 4 27 & y^3-y^2+ 4 27 =0 & 27 y^3-27 y^2+4=0 aligned This is a cubic equation. If we put y=- 1 3 then (3 y+1)=0 is a factor of cubic equation. aligned & (3 y+1) (9 y^2-12 y+4 )=0 & (3 y+1)(3 y-2)^2=0 aligned Hence y=- 1 3 , 2 3 Thus f(x)=1 has two solutions.