Question

Graph the function and identify all key characteristics f(x)=x^3-12x^2+45x-54

Answer 1

The key characteristics of the function are

Domain: (-∝, ∝)

Range: (-∝, ∝)

Relative Maximum: (3, 0)

Relative Minimum: (5, -4)

End behavior: As x ⇒ +∝, f(x) ⇒ +∝ and As x ⇒ -∝, f(x) ⇒ -∝

Increasing interval = (-∝, 3) and (5, ∝)

Decreasing interval = (3, 5)

Zeros = (3, 0) and (5, 0)

Identify all key characteristics of the function

From the question, we have the following parameters that can be used in our computation:

f(x) = x³ - 12x² + 45x - 54

The graph of the function f(x) is plotted and attached

From the graph, we have the following readings:

Domain: (-∝, ∝)

This is because the x values can assume any real value

Range: (-∝, ∝)

This is because the y values can also assume any real value

Relative Maximum: (3, 0)

Relative Minimum: (5, -4)

Also, we have the increasing interval to be (-∝, 3) and (5, ∝)

This is because the y values increase as x increase

And the decreasing interval is (3, 5)

This is because the y values decrease as x increase

The end behavior is

As x ⇒ +∝, f(x) ⇒ +∝

As x ⇒ -∝, f(x) ⇒ -∝

Lastly, the zeros of the graph are (3, 0) and (5, 0)

This is because the graph intersect the x-axis at these points

Answer 2

We kindly ask you to see the explanation and the image attached below for further details.

Explanation:

From Function Theory we know that third order polynomials are continuous functions with one relative minimum and one relative maximum and whose domain and range comprehends the set of all real numbers. Since the leading coefficient of the polynomial is positive, then two increasing intervals and only one increasing interval exist.

Now we proceed to graph the function and label each relevant information when possible.

Domain

`Dom \{f(x)\} = \mathbb{R}`

Range

`Ran \{f(x)\} = \mathbb{R}`

Zeros

x = 3 and x = 6