If the function f(x)=a x^3+b x^2+26 x-24 satisfies the conditions of Rolle's theorem in [2,4] and f^ (3+ 1 3 )=0 , then the value of a b is equal to

Step-by-step solution

f(x)=a x^3+b x^2+26 x-24 ...(i) on [2,4] f(x) satisfies the Role's theorem array ll & f(2)=f(4) & a(2)^3+b (2^2 )+26(2)-24 array =a(4)^3+b(4)^2+26(4)-24 8 a+4 b+28=64 a+16 b+80 56 a+12 b+52=014 a+3 b+13=0 ...(ii) On differentiating Eq. (i) w.r.t. x aligned & f^ (x)= d d x (a x^3+b x^2+26 x-24 ) & f^ (x)=3 a x^2+2 b x+26 aligned At, x=3+ 1 3 f^ (3+ 1 3 )=3 a (3+ 1 3 )^2+2 b (3+ 1 3 )+26 aligned & 0=3 a (9+ 1 3 + 6 3 )+6 b+ 2 b 3 +26 & 0=28 a+6 b+6 3 a+ 2 b 3 +26 aligned (28+6 3 ) a+ ( 6 3 +2 3 ) b+26=0 ...(iii) (28