Answer:
Yes, this equation has a solution. According to Intermediate Value Theorem at least one solution for [0,2]
Step-by-step explanation:
Hi there!
1) Remember a definition.
Intermediate Value Theorem:
If [tex]f[/tex] is continuous on a given closed interval [a,b], and f(a)≠f(b) and f(a)<k<f(b) then there has to be at least one number 'c' between 'a' and 'b', such that f(c)=k
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(Check the first graph as an example)
2) The Intermediate Value Theorem can be applied to determine whether there is a solution on a given interval.
Let's choose the interval [tex][0,2][/tex]
[tex]f(x)=x^{3}-3x-1\\f(0)=(0)^{3}-3(0)-1\\f(0)=-1\\f(0)<0\\[/tex]
Proceed to the other point: 2
[tex]f(x)=x^{3}-3x-1\\f(2)=(2)^{3}-3(2)-1\\f(2)=1\\f(2)>0\\[/tex]
3) Check the 2nd Graph for a the Visual answer, of it. And the 3rd graph for all solutions of this equation.
