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Question

If the function f given by
f(x)=x33(a2)x2+3ax+7, for some aR is increasing in (0,1] and decreasing in [1,5), then a root of the equation,
f(x)14(x1)2=0 (x1) is :

A
7
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B
5
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C
6
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D
7
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Solution

The correct option is D 7
f(x)=x33(a2)x2+3ax+7
f(x)=3x26(a2)x+3a
f(x)0 x(0,1]
f(x)0 x[1,5)
x=1 is a critical point.
f(1)=0
36a+12+3a=0
​​​​​​​a=5
​​​​​​​
​​​​​​​f(x)14(x1)2=0
​​​​​​​x39x2+15x7(x1)2=0
(x1)2(x7)(x1)2=0
​​​​​​​x=7 is a root.

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