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I am suppose to use Rolle's Theorem and then find all numbers c that satisfy the conclusion of the theorem.

$f(x) = x^4 +4x^2 + 1 [-3,3]$

Polynomials are always going to satisfy the theorem.

The derivative is

$4x^3 + 8x$ and the only number that could possibly make that zero would be zero so the answer is 0.

What did I get wrong? I failed my calc test and this was one of the one's I got wrong.

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    You have, for $a = -3, b = 3$, $f(a) = f(b)$. By the theorem, there exists $c \in (-3, 3)$ such that $f \ '(c) = 0$. Solving $$4x^3 + 8x = 0$$ gives $x = 0$ as the only real solution... am I missing something too?? [Note - polynomials will always satisfy *the continuous and differentiable conditions*]2012-04-17
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    In general, you should ask your instructor what went wrong before other people. There are many reasons an answer might not be totally correct, and the instructor is the only one who can tell you exactly why you are wrong. That being said, Kannapan's answer would be my guess as to where you went wrong.2012-04-17
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    The instructor said that I didn't show enough work to prove that 4x^3 + 8x = 0, I am planning to petition the result I got on the test so I just wanted some clarification.2012-04-17
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    Perhaps they were looking for you to explain that $4x^2+8>0$ for all $x$?2012-04-17

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