Finding the number of solutions of an equation

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Homework Help Overview

The problem involves analyzing the function \( h_k(x) = x^3 - 6x + k \) to determine the values of \( k \) that yield one, two, or three solutions to the equation \( h_k(x) = 0 \). The subject area pertains to polynomial functions and their roots.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss graphing the function to visualize the number of solutions and consider the behavior of the function at specific values of \( k \). Questions arise about how to systematically determine the number of solutions based on the graph's intersections with the x-axis.

Discussion Status

Some participants have provided suggestions for graphing and analyzing the function, while others are exploring how changes in \( k \) affect the number of solutions. There is an ongoing exploration of different values of \( k \) and their implications for the function's roots.

Contextual Notes

Participants note that the value of \( k \) can vary, and there is a reference to previous values of \( k \) used in earlier discussions. The original poster expresses uncertainty about the approach to take.

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Homework Statement


Let hk : R -> R whereh_{k}(x)=x^3-6x+k and k is a real number

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

The Attempt at a Solution



Don't know how to approach this.

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

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