If f(x)=x^3+p x^2+q x is defined on [0,2] such that f(0)=f(2) and f^ (1+ 1 3 )=0 , then p^2+q^2=

Step-by-step solution

Given, f(x)=x^3+p x^2+q x is defined on [0,2] array ll f(0)=f(2) & & 0=2^3+p(2)^2+q(2) & 4 p+2 q+8=0 array aligned Now f^ (x) & = d d x (x^3+p x^2+q x ) f^ (x) & =3 x^2+2 p x+q aligned Given, f^ (1+ 1 3 )=0 aligned & 3 (1+ 1 3 )^2+2 p (1+ 1 3 )+q=0 & 2 p (1+ 1 3 )+q+3 (1+ 1 3 + 2 3 )=0 aligned Eq. (ii) Subtracting by Eq. (i), we get aligned & 2 p 3 + 6 3 =0 & 2 P+6=0 P=-3 aligned Substituting P=-3 in Eq. (i) aligned & 2(-3)+q+4=0 & q=2 & p^2+q^2=(-3)^2+(2)^2=9+4 & p^2+q^2=13 aligned Hence, option (a) is correct.