If p(x) be a polynomial of degree 3 satisfying p(−1) = 10, p(1) = −6 and p(x) has maximum at x = −1 and p′(x) has minima at x = 1. Find the distance between the local maximum and local minimum of the curve.

Hello Student,

Let the polynomial be p(x)=ax^3 +bx^2 +cx+d
p(-1)=–a + b – c + d = 10
p (1) = a + b + c + d = –6
p'(-1)=3a – 2b + c = 0
p''(1)= 6a + 2b = 0
a,b,c,d=1,-3,-6,5
p'(x)=3(x+1)(x–3)
x=-1 max
x=3 min
distance =(-1,10) and (3,-22) is 4 sqrt{65}

Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty