Question

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

Answer 1

See explanation.

Explanation:

`y =f(x)=x^3-12x^2+45x-54`, with a zero x = 3.

`y'=3x^2-24x+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(112/x+45/x^2-64/x^3) to +-oo`, as `x to +-oo`.

Domain and range: `(- oo, oo)`

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