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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f(x) be a monic polynomial of degree 4 satisfying the following conditions: (i) f^ (0)=0 (ii) f^ (2)=16 (iii) for some positive real k, f^ (x) < 0 in the intervals (- , 0) and (0, k) . Then which of the followings is/are correct?

Options

  1. AEquation f^ (x)=0 has one real root in the interval (0,2) .
  2. BFunction f(x) has a local maximum.
  3. CIf f(0)=0 , then for all reals x, f(x) - 1 3 .
  4. DIf f(0)=0 , then the value of | _ -1 ^1 f^ (x) 4 d x |= 2 3 .

Correct answer

D. If f(0)=0 , then the value of | _ -1 ^1 f^ (x) 4 d x |= 2 3 .

Step-by-step solution

Since, f^ (0)=0 and f(x) decreases in the intervals (- , 0) and (0, k) , we infer that it contacts the line y=f(0) as it passes through Then, f(x) would look like the picture shown on the right. Since f^ (2)>0 , we can notice that k < 2 , and therefore f^ (x)=0 has one root in the open interval (0,2),(a , true) and obviously f(x) does not have a local maximum ( b , false). If f(0)=0 , then f(x)=x^3(x-p) . This leads to f^ (x)=3 x^2(x-p)+x^3=x^2(4 x-3 p) and from f^ (2)=16 , we get p= 4 3 . f^ (x)=4 x^2(x-1) , and t

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