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Question

If f(x)=x3+4x2+kx+1 is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then .


Correct Answer
A
k = 4
Your Answer
B
k = 2
Your Answer
C
k = -1
Your Answer
D
k has no real value 

Solution

The correct option is A k = 4
Here,f'(x)0or, 3x2+8x+k0,  x(2, 23)
Since, 3x2+8x+k represents a upward parabola. Then , the situation for f '(x) are as follows:
Case1.

Case2.

Given that f(x) decreases in the largest possible interval (2,23). Then,f '(x) must have roots -2 and -2/3.
Thus, Product of roots= (2) (23) =k3or, k=4.

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