Derivatives effect on graph?

Question

sketch the graph of the function

x^4-x^3

 

i was wondering on how do we get the points that we have to plug on the graph? Are they only the x and y intercepts?

Answer 1

PROBLEM

 

Derivatives effect on graph?
sketch the graph of the function
x^4-x^3

i was wondering on how do we get the points that we have to plug on the graph? Are they only the x and y intercepts?

 

SOLUTION

 

The function f(x) = x⁴ - x³ has four roots (zeros) and is an even function, which means it essentially looks something like this:

 

♥*****************♥

*******************

*♥*******♥*******♥*

**♥***♥*****♥***♥**

****♥*********♥****

 

We know that the end behavior is x→-∞, f(x)→∞ and x→∞, f(x)→∞

 

If you factor the function, you get f(x) = x³(x - 1).

 

Therefore, f(x) = 0 at x = 0 and x = 1. So, there are three zeros at x = 0 and one zero at x = 1.

 

To find the minimum value, take the derivative of f(x) and set it equal to zero.

 

So, f'(x) = 0 = 4x³ - 3x²

 

4x³ = 3x²

 

4x³ = 3x²

----     ----

 x²       x²

 

4x = 3

 

 x = 3/4

 

f(3/4) = (3/4)³(3/4 - 1)

 

f(3/4) = (27/64)(-1/4)

 

f(3/4) = -27/256 = -0.10546875

 

You can now plot a few points to fill in your graph...

 

     x      |      f(x)    

    -2      |      24     

    -1      |       2      

     0      |       0      

   1/4     |  -3/256   

   1/2     | -16/256  

   3/4     | -27/256  
     1      |       0      
     2      |       8      
     3      |      54     

Now, graph the function...

 

♥*********************♥(3,54)******

♥*********************♥***********

♥********************♥************

♥(-2,24)**************♥************

♥*******************♥*************
♥******************♥(2,8)**********

*♥****************♥***************
**♥(-1,2)*********♥****************
----♥(0,0)♥♥-------♥(1,0)--------------------
*********♥(1/4,-3/256)**************

**********♥(1/2,-16/256)************

***********♥(3/4,-27/256)***********