Find the maximum and minimum value, f(x) = 9x^2 + 12x + 2 - Vidyadip

Question

Find the maximum and minimum value, $f(x) = 9x^2 + 12x + 2$

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Answer

It is given that $f(x) = 9x^2 + 12x + 2 = (3x + 2)^2 - 2$
Now, we can see that $(3x + 2)^2 \ge 0$ for every $x \in R$
$\Rightarrow f(x) = (3x + 2)^2 - 2 \ge -2$ for every $x \in R$
The minimum value of $f$ is attained when $3x + 2 = 0$
$3x + 2 = 0$
$\Rightarrow \mathrm{x}=-\frac{2}{3}$
Then, Minimum value of $f=f\left(-\frac{2}{3}\right)=\left(3\left(-\frac{2}{3}\right)+2\right)^{2}-2=-2$
Also, since $x = -\frac{2}{3}$, is the only critical point which is a minimum,
Therefore, function $f$ does not have a maximum value.

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