Chapter 2.2, Problem 85E

To determine

To draw: the graph of given function $f(x)=x^3-3x^2$

Answer to Problem 85E

  

Explanation of Solution

Given information:

Function $f(x)=x^3-3x^2$ and conditions are

1. by applying the leading coefficient test;
2. finding the zeros of polynomial;
3. plotting sufficient solution;
4. Draw a continuous curve through the points.

Calculation:-

Step: - (a) − Applying the leading coefficient test, we have

In given function, the leading coefficient is +1 (positive) and power of x is 3 (odd) so the graph will drop down to the left and will rise up to right then graph will be something like this

  

Step (b):- now find the zeros of given function, then

  $\begin{array}{l}{x}^3-3x^2\\ =x^2(x-3)\end{array}$

Real Zeros are 0, 0, 3

Step (c):- since zeros of an equation satisfies the equation so, f(x) = 0 at x = 0 and 3.

Also the multiplicity of zero (0) is 2 i.e. multiplicity is even so the function will touch the x-axis at x = 0 (if this is odd i.e. 3 or 5 then function will cross the x-axis).

Then graph will be something like this

  

Step (d):- now drawing the continuous curve by using step (a), (c). We have