Prove that the function does not have maxima or minima: h(x) = x^… - Vidyadip

Question

Prove that the function does not have maxima or minima: $h(x) = x^3 + x^2 + x +1$

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Answer

$h(x) = x^3 + x^2 + x + 1$
$\Rightarrow h'(x) = 3x^2 + 2x + 1$
$h(x) = 0$
$\Rightarrow 3x^2 + 2x + 1 = 0$
$\Rightarrow \mathrm{x}=\frac{-2 \pm 2 \sqrt{2} \mathrm{i}}{6}$
$\Rightarrow \frac{-1 \pm \sqrt{2} i}{3}, \notin R$
Therefore, there does not exist $ C \in R$ such that $h'(c) = 0,$ i.e, there are no real critical points.
Hence, function h does not have maxima or minima.

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