f ( x ) is differentiable on R and f ^ ( m ) 0, ~m R . If _ x m x f(m)-m f(x) x-m +f^ (m)=f(m) , then m=

Step-by-step solution

Let f(x)= 1 3 x^3+2 x f^ (x)=x^2+2 0 x R Now, _ x m x f(m)-m f(x) x-m +f^ (m)=f(m) aligned & _ x m x ( 1 3 m^3+2 m )-m ( 1 3 x^3+2 x ) x-m +m^2+2 & = 1 3 m^3+2 m & _ x m - 1 3 m x(m+x)(m-x) (m-x) +m^2+2= 1 3 m^3+2 m & - 2 m^3 3 -m^2+2= 1 3 m^3+2 m & m^3-m^2+2 m-2=0 & (m-1) (m^2+2 )=0 m=1 aligned