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Question

# If the equation x3โ9x2+24x+k=0 has exactly one root in (2,4), then k lies in the interval

A
(20,16)
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B
(16, )
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C
(,20)
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D
None of these
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Solution

## The correct option is A (−20,−16)Let f(x)=x3−9x2+24x+k=0If f has exactly one root in (a,b), then f(a)⋅f(b)<0For (a,b)≡(2,4), we can say⇒f(2)⋅f(4)<0⇒(8−9(4)+24(2)+k)(64−9(16)+24(4)+k)<0⇒(k+20)(k+16)<0⇒k∈(−20,−16)f′(x)=3(x−2)(x−4) Hence in the interval (2,4),f′(x) is negative. Hence, it is monotonic decreasing in that interval and can have only one real root.Hence, option A.

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