Prove that the following function does not have maxima or minima.

$h (x) = x^{3} + x^{2} + x + 1$

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Solution

Given function is, $h (x) = x^{3} + x^{2} + x + 1$
$\Rightarrow h^{'} (x) = 3 x^{2} + 2 x + 1$
Now, put $h^{'} (x) = 0 \Rightarrow 3 x^{2} + 2 x + 1 = 0$
$\Rightarrow x = \frac{- 2 \pm \sqrt{4 - 4 \times 3 \times 1}}{6} = \frac{- 2 \pm \sqrt{- 8}}{6} [U s i n g x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}] \Rightarrow x = \frac{- 2 \pm 2 \sqrt{2} i}{6} = \frac{2 (- 1 \pm \sqrt{2} i)}{6} = \frac{- 1 \pm \sqrt{2} i}{3} / ϵ R$
Therefore, there does not exist any $x ϵ R$ such that h' (x)=0
Hence, function h does not have maxima or minima

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