Prove that the function does not have maxima or minima: h(x) = x3… - Vidyadip

Question

Prove that the function does not have maxima or minima: h(x) = x³ + x² + x +1

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Answer

h(x) = x³ + x² + x + 1
$\Rightarrow$ h'(x) = 3x² + 2x + 1
h(x) = 0
$\Rightarrow$ 3x² + 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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