Finding the number of solutions of an equation

[Rudders]

Homework Statement

Let h_k : R -> R where$$h_{k}(x)=x^3-6x+k$$ and k is a real number

Find the value(s) of k for which the equation h_k has 1,2 or 3 solutions.

The Attempt at a Solution

Don't know how to approach this.

Thanks!

 

[rootX]

Can you graph the function?
Find max min and do that. It should help.

 

[Rudders]

Hi,

Yes, i have graphed the function with the value of k being -8,0,6 and 10 (that was a previous question), but the value k can be anything for this question. How can I find the number of solutions produced?

 

[HallsofIvy]

How many times does the graph cross the x-axis? That's the number of solutions.

Obviously, if k= 0 then the function (not equation) is $h_0(x)= x^3- 6x$ and presumably you are talking about the equation $h_0(x)= x^3- 6x= x(x^2- 6)= 0$ which has 3 solutions, 0 and $\pm\sqrt{6}$. Now try a few other values of k and see how the graph changes and how many times the graph crosses the x-axis.