If f(x)=p x^3+q x^2+r x+t attains local minimum and local maximum values at x=-2 and x=2 respectively and p is a root of 9 x^2-1=0 , then p + q + r =

Step-by-step solution

f^ (x)=3 p x^2+2 q x+r ...(i) Since, f(x) attain local minimum and maximum at x=-2 and x=2 respectively. f^ (x)=k(x+2)(x-2)=k (x^2-4 ) ...(ii) From (i) and (ii) aligned & k (x^2-4 )=3 p x^2+2 q x+r & k x^2-4 k=3 p x^2+2 q x+r aligned Comparing both sides, we get k=3 p, q=0 and r=-4 k Now, f^ (x)=6 p x+2 q=6 p x q=0 f^ (-2)=-12 p>0 x=-2 is local minimum p < 0 p is a root of 9 x^2-1=0 x= 1 3 p < 0 p=- 1 3 So, k=3 (- 1 3 )=-1 r=-4 k=-4(-1)=4 p+q+r=- 1 3 +0+4= 11 3 .