The minimum value of $e^{(x^{4} - x^{3} + x^{2})}$ is

e

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e^{2}

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1

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None of these

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Solution

The correct option is C $1$
$e^{(x^{4} - x^{3} + x^{2})}$ is minimum when $(x^{4} - x^{3} + x^{2})$ is minimum
since $x^{4} - x^{3} + x^{2} = x^{2} (x^{2} - x + 1)$
$= x^{2} [{(x - \frac{1}{2})}^{2} + \frac{3}{4}] \geq 0, \forall x \in R$
$∴$ the minimum value of $x^{4} - x^{3} + x^{2}$ is $x = 0$
Hence the minimum value of $e^{(x^{4} - x^{3} + x^{2})}$ is $e^{0} = 1$

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[MNR 1990; AMU 1999; UPSEAT 2000; Pb. CET 2004]

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The minimum value of e(x4−x3+x2) is

The correct option is C 1e(x4−x3+x2) is minimum when (x4−x3+x2) is minimumsince x4−x3+x2=x2(x2−x+1)=x2[(x−12)2+34]≥0,∀x∈R∴ the minimum value of x4−x3+x2 is x=0Hence the minimum value of e(x4−x3+x2) is e0=1

The minimum value of $e^{(x^{4} - x^{3} + x^{2})}$ is