# How do you find the max or minimum of f(x)=3x^2-7x+2?

Dec 28, 2016

There is a minimum at $\left(\frac{7}{6} , - \frac{25}{12}\right)$

#### Explanation:

We can factorise $f \left(x\right) = \left(3 x - 1\right) \left(x - 2\right)$

We calculate the derivative of $f \left(x\right)$ and any critical point is found when $f ' \left(x\right) = 0$

$f \left(x\right) = 3 {x}^{2} - 7 x + 2$

$f ' \left(x\right) = 6 x - 7$

$f ' \left(x\right) = 0$, when $6 x - 7 = 0$. $\implies$, $x = \frac{7}{6}$

We make a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a}$$\frac{1}{3}$$\textcolor{w h i t e}{a a a a a}$$\frac{7}{6}$$\textcolor{w h i t e}{a a a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$f ' \left(x\right)$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a}$color(white)(aaa)↘$\textcolor{w h i t e}{a a}$$- \frac{25}{12}$$\textcolor{w h i t e}{a a a a a a}$↗$\textcolor{w h i t e}{a a}$

There is a minimum at $\left(\frac{7}{6} , - \frac{25}{12}\right)$

We can also calculate $f ' ' \left(x\right) = 6$

$f ' ' \left(x\right) > 0$, we have a minimum