If f(x)=x3+4x2+kx+1 is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then .
A
k = 4
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B
k = 2
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C
k = -1
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D
k has no real value
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Solution
The correct option is A k = 4 Here,f'(x)≤0or,3x2+8x+k≤0,∀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,Productofroots=(−2)(−23)=k3or,k=4.