# How do you graph f(x)=x^3-12x^2+45x-54 and identify domain, range, max, min, end behavior, zeros?

Dec 7, 2016

See explanation.

#### Explanation:

$y = f \left(x\right) = {x}^{3} - 12 {x}^{2} + 45 x - 54$, with a zero x = 3.

$y ' = 3 {x}^{2} - 24 x + 45$, with zeros x = 3 and 5.

x = 3 is a double root. So, the third root is 9.

y'' = 6x - 24 = 0, at x = 4, < 0 at x = 3 and > 0 at x = 5..

y'''= 6.

Maximum y = y(3) = 0.

Minimum y = y(5) = $- 4$.

Point of inflexion: (4. -2).

$y = {x}^{3} \left(\frac{112}{x} + \frac{45}{x} ^ 2 - \frac{64}{x} ^ 3\right) \to \pm \infty$, as $x \to \pm \infty$.

Domain and range: $\left(- \infty , \infty\right)$

graph{x^3-12x^2+45x-54 [-10, 10, -5, 5]}