Class 11Mathematics4-2.Quadratic Equations and InequationsSolution of quadratic equations and Nature of roots
MediumFind the maximum and minimum values of the function $f(x) = 9x^{2} + 12x + 2$.
(D) The given function is $f(x) = 9x^{2} + 12x + 2$.
We can rewrite this by completing the square:
$f(x) = (3x)^{2} + 2(3x)(2) + 2^{2} - 2^{2} + 2$
$f(x) = (3x + 2)^{2} - 4 + 2$
$f(x) = (3x + 2)^{2} - 2$.
Since $(3x + 2)^{2} \geq 0$ for all $x \in \mathbb{R}$,the minimum value of $(3x + 2)^{2}$ is $0$.
Therefore,the minimum value of $f(x)$ is $0 - 2 = -2$,which occurs when $3x + 2 = 0$,i.e.,$x = -\frac{2}{3}$.
As $x \to \infty$,$f(x) \to \infty$,so the function does not have a maximum value.
We can rewrite this by completing the square:
$f(x) = (3x)^{2} + 2(3x)(2) + 2^{2} - 2^{2} + 2$
$f(x) = (3x + 2)^{2} - 4 + 2$
$f(x) = (3x + 2)^{2} - 2$.
Since $(3x + 2)^{2} \geq 0$ for all $x \in \mathbb{R}$,the minimum value of $(3x + 2)^{2}$ is $0$.
Therefore,the minimum value of $f(x)$ is $0 - 2 = -2$,which occurs when $3x + 2 = 0$,i.e.,$x = -\frac{2}{3}$.
As $x \to \infty$,$f(x) \to \infty$,so the function does not have a maximum value.
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