[Math] Using Rolle’s Theorem find numbers

calculus

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.

Best Answer

Polynomials are always going to satisfy theorem.

No. They satisy the hypothesis about continuity and differentiability in the closed and open intervals respectively while you need to check if the polynomials agree at the end point.

Here they will agree on the end points because $x^2$ and $x^4$ are even functions and the interval is symmetric about $0$. (Do you see why?)

And, you are in need of $c$ that satisfies $f'(c)=0$. You're right but I would be surprised if the instructor does not want you to prove your claim:

\begin{align} f'(c)&=0 \;\;\mbox{and $c \in (-3,3)$} \tag{1}\\4c^3+8c&=0\\4c(c^2+2)&=0 \end{align}

Since $c^2+2 \ge 2$ for $c \in \Bbb R$, the equation $c^2+2=0$ does not have roots in $\Bbb R$ and hence $c=0$ is the only value that satisfies $(1)$.

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