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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Consider a polynomial P(x) of the least degree that has a maximum equal to 6 at x=1 , and a minimum equal to 2 at x=3 . Let P(2)+P^ (0) is equal to n then [ n 2 ] is

Correct answer

6

Step-by-step solution

The polynomial is an everywhere differentiable function. Therefore, the points of extremum can only be roots of the derivative. Furthermore, the derivative of a polynomial is a polynomial. The polynomial of the least degree with roots x₁=1 and x₂=3 has the from a(x-1)(x-3) Hence P^ (x)=a(x-1)(x-3)=a (x^2-4 x+3 ) since at the point x=1 , there must be P(1)=6 , we have P(x)= ₁^x P^ (x) d x+6=a ₁^x (x^2-4 x+3 ) d x+6=a ( x^3 3 -2 x^2+3 x- 4 3 )+6 The coefficient ' a ' is determined from the condition P(3)=2 , whence a

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