Let p (x) be a polynomial of degree three that has a local maximum value 8 at x =1. and a local minimum value 4 at x = 2, then p (0) is equal to

Let p (x) be a polynomial of degree three that has a local maximum value 8 at x =1. and a local minimum value 4 at x = 2, then p (0) is equal to

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    Solution:

    We find that p' (x) is a quadratic polynomial having x = 1 and x = 2 as its two zeros.

               p(x)=λ(x1)(x2).where λ is a positive constant.

     p(x)=λx3332x2+2x+C                       ..(i)

    where C is a constant .

    It is given that p (1) = 8 and p (2) = 4.

        λ1332+2+C=8 and λ836+4+C=4    5λ6+C=8 and 2λ3+C4C=12 and λ=24

    Hence, p(0)=C=12.

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