Spreadsheet of

 


Equation?

Can you prove that the equation has at most one solution? Of course.

We can use the intermediate value theorem to prove it. Look around my proof.

 

Let f(x)=.

Then f is continuous because f(x) is a polynomial, and f(0)=1>0, while f(-1)=-2<0.

So by the intermediate value theorem, f(c)=0 for some number c in (-1,0).

So the equation =0 has at least one solution. Also f '(x)=.

Because f '(x)>0 for all x=~0, and f '(0)=0, f is increasing on R.

Therefore, f (x)=0 has at most one solution.


 

I would like to check my proof using the spreadsheet of Excel. The spreadsheet will show the value of y in the equation with the various value of x. First of all, check the value of x, [1,20].

-5 -81374
-4 -17471
-3 -2456
-2 -167
-1 -2
0 1
1 4
2 169
3 2458
4 17473
5 81376

 


Let's check that the equation f(c)=0 for some numver c in (-1, 0). The following is the spreadsheet in [-5,5]. We will know that the y-value changes from negative to positive in [-1,0].

 

-5 -81380
-4 -17476
-3 -2460
-2 -170
-1 -4
0 0
1 4
2 170
3 2460
4 17476
5 81380

 


Using the spreadsheet, we can easily find the y-value of the function, and imagine the graph of the function. In the mathematics class, spreadsheet can supply students the material, with which they can check their idea and imagine the other steps.


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