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Question

Let f(x)=x4λx33x2+3xλxλ, xR{λ}. If the range of f(x) is R, then the complete set of values of λ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

A
[2,2]
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B
[0,4]
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C
(1,3)
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D
(2,2)
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Solution

The correct option is A [2,2]
Given : f(x)=x4λx33x2+3xλxλ
f(x)=(x33x)(xλ)xλf(x)=x(x23)(xλ)xλ

Considering g(x)=x33x


f(x)=g(x) except x=λ
If λ>2, then from the range of f(x), f(λ) is missed,
Similarly, if λ<2, then from the range of f(x), f(λ) is missed,
But when λ[2,2], f(λ) is achieved at some other point than x=λ

Hence, λ[2,2]

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