1. Intermedaite value theorm

given $f(x)=x^4+3x^2-x+1$ show that there exist at least three real numbers x such that $f(x)=3$

how could I prove that there is at least three?

2. Originally Posted by akhayoon
given $f(x)=x^4+3x^2-x+1$ show that there exist at least three real numbers x such that $f(x)=3$

how could I prove that there is at least three?
this is not true. there are only two. check to make sure you typed the right problem

3. rechecked confirmed, this is how the quiz question was written.

4. Originally Posted by akhayoon
rechecked confirmed, this is how the quiz question was written.
Below are the graphs of y = x^4 + 3x^2 - x + 1 and y = 3. as you can clearly see, they intersect only twice. namely at x = -0.6337532429 and x = 0.8738358889. thus there is something wrong with the question.

as you can see, the graph looks like a parabola, so, in fact, f(x) = c at most twice for any c

5. Originally Posted by Jhevon
this is not true. there are only two. check to make sure you typed the right problem
I agree with Jhevon. (And so does my calculator. )

-Dan

6. ok, so would I use the I.V.T to prove them wrong?

7. Originally Posted by akhayoon
ok, so would I use the I.V.T to prove them wrong?
the question does not have a "prove or disprove" instruction, so i would not really recommend it. just tell your professor that there is something wrong with the question.

you can use IVT to prove there is at least 2. to disprove that there are at least 3 cannot be done with IVT as far as i can see, but we can disprove it (if that's allowed) using Rolle's theorem

8. Originally Posted by akhayoon
given $f(x)=x^4+3x^2-x+1$ show that there exist at least three real numbers x such that $f(x)=3$

how could I prove that there is at least three?
So the nearest correct problem might be

Given f(x)=x^3+3x^2-x+1, then there exist at least three real numbers x such that f(x)=3 (See the graph below).