Intermediate-value theorem

[katem]

I have a final coming up and I was reviewing old exams, I have a question on one of the problems, which is,
"Give that f(x)=x⁴-x²+5x+2, Use the intermediate-value theorem to show that there exists a real number c such that f(c)=3. Explain why the intermediate value theorem can be used."
Learning how to use the theorem we were always given an interval that the solution was found between so I'm not sure how to go about setting this problem up.
Any and all help would be greatly appreciated!
Thank you!

 

[pka]

I have a final coming up and I was reviewing old exams, I have a question on one of the problems, which is,
"Give that f(x)=x⁴-x²+5x+2, Use the intermediate-value theorem to show that there exists a real number c such that f(c)=3. Explain why the intermediate value theorem can be used."
Learning how to use the theorem we were always given an interval that the solution was found between so I'm not sure how to go about setting this problem up.

What are the values of $\displaystyle f(0)~\&~f(1)~?$. Is that your interval?

 

[stapel]

I have a final coming up and I was reviewing old exams, I have a question on one of the problems, which is,

"Give that f(x)=x⁴-x²+5x+2, Use the intermediate-value theorem to show that there exists a real number c such that f(c)=3. Explain why the intermediate value theorem can be used."

Learning how to use the theorem we were always given an interval that the solution was found between so I'm not sure how to go about setting this problem up. Any and all help would be greatly appreciated! Thank you!

What, exactly, does the Intermediate-Value Theorem say? How have you related (or how do you think you can relate) the given function to this Theorem?

Please be complete. Thank you!

 

[katem]

What, exactly, does the Intermediate-Value Theorem say? How have you related (or how do you think you can relate) the given function to this Theorem?

Please be complete. Thank you!

If x is continuous on [a,b] and N is any number between f(a) and f(b) then there is at least one number c in the interval (a,b), such that f(c)=N
so in this case N is 3. So do I just set f(c) equal to 3 and solve? The function is continues from negative to positive infinity because it is a polynomial but how does that relate to the theorem?
Thanks for the help i'm sure this is really basic so sorry for that.

 

[katem]

What are the values of $\displaystyle f(0)~\&~f(1)~?$. Is that your interval?

Why did you choose 0 and 1?

 

[HallsofIvy]

Why did you choose 0 and 1?

He chose them because they worked! Sometimes you have to try things until you find something that works. In order to find an interval in which there must be x such that f(x)= 3, you need to find value of x so that f(x)< 3 and another so that f(x)> 3. The simplest thing to do is try integers. If you had just started evaluating f at simple values, trying to find an interval that works, originally, you could have done the problem yesterday!