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Question

f(x)=x33+x22+ax+bxR, then find least value of 'a' for which f(x) is injective function

A
14
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B
1
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C
12
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D
18
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Solution

The correct option is B 14
given function is f(x)=x33+x22+ax+b,xϵR and alsof(x) is injective.
f(x) should be strictly increasing or decreasing function.
1) f(x) is a strictly increasing function
f(x)>0
x2+x+a>0
For a quadratic equation to be positive the coeffcient of x2 should be positive and discriminant of the quadratic equation should be negitive
Here coefficient of x2 is positive.
Δ<0
14a<0
a>14
the minimum value of a is 14.
2)f(x) is strictly decreasing function
f(x)<0

x2+x+a<0
A quadratic equation can be negitive only if the coeffcient of x2 is negitive,but here it is positive
this is not a case
Hence the minimum value of a is 14.

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