QuestionGiven a continuous function $f$ on $[-2,2]$ with $f(-2)=1$ and $f(2)=-1$, which properties hold by the Intermediate Value Theorem? A. $f(c)=0$ for some $c$ in $(-1,1)$; B. $f(x)+1 \geq 0$ on $(-2,2)$; C. $f^{2}(c) \leq 1$ for all $c$ in $(-2,2)$.

Studdy Solution
To check option C, we need to see if $f^{2}(c) \leq1$ for all $c$ in $(-2,2)$. Given that $f(-2)=1$ and $f(2)=-1$, we know that $f$ takes on values between $-1$ and $1$ in the interval $(-2,2)$. Therefore, $f^{2}(c)$ will always be less than or equal to $1$ for all $c$ in $(-2,2)$.
So, the only property that follows without further restriction on $f$ by applying the Intermediate Value Theorem is C.