Show that the function $f (x) = x^{2} + 3 x^{2} + 3 x + 107$ is increasing for all $R$ .

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Solution

$f (x) = x^{3} + 3 x^{2} + 3 x + 107$

$f (x) = 3 x^{2} + 6 x + 3$

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

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

since, $f^{'} (x) > 0$

$\forall x ϵ R$

Hence $f (x)$ is increasing

for all R.

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