Find the maximum and minimum value, f(x) = 9x2 + 12x + 2 - Vidyadip

Question

Find the maximum and minimum value, f(x) = 9x² + 12x + 2

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Answer

It is given that f(x) = 9x² + 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 = -$\frac23$, is the only critical point which is a minimum,
Therefore, function f does not have a maximum value.

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