Minimum value of function f ( x ) = x 3 ( x 3 + 1 ) ( x 3 + 2 ) ( x 3 + 3 ) : x ∈ R , is

Let t = x 3 ( x 3 + 3 ) ; t = ( x 3 + 2 3 ) 2 − 4 9 ∈ [ − 4 9 , ∞ ) f ( x ) = g ( t ) = t ( t + 2 ) = ( t + 1 ) 2 − 1 is least when t = − 1 and − 1 ∈ [ − 4 9 , ∞ ) ∴ min f ( x ) = − 1

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MathematicsFunctionsDomain-RangeMedium2 minQB

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Minimum value of function $f (x) = x^{3} (x^{3} + 1) (x^{3} + 2) (x^{3} + 3) : x \in R$ , is

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Answer:

Solution:

Let $t = x^{3} (x^{3} + 3); t = (x^{3} + \frac{3}{2})^{2} - \frac{9}{4} \in [- \frac{9}{4}, \infty)$

$f (x) = g (t) = t (t + 2) = (t + 1)^{2} - 1$ is least when $t = - 1$

and $- 1 \in [- \frac{9}{4}, \infty) ∴ min f (x) = - 1$

Stream:JEESubject:MathematicsTopic:FunctionsSubtopic:Domain-Range

⏱ 2mℹ️ Source: QB

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