I have discovered a series of calculus questions. However, I am not really equipped with the necessary knowledge to answer them. How should theorems like the IVP, mean value theorem, etc. be applied to answer these questions?
State whether the given situation is possible or impossible
Consider the interval to be $[a,b]$, $a>0$, and that $c$ is an element of interval $[a,b]$. Suppose also that each function $f$ is continuous over $[a,b]$:
$f$ has a unique positive maximum, a unique negative minimum and two values $c$ such that $f(c)=0$
$f$ has two maxima, two minima, and a unique value $c$ such that $f(c)=0$
$f$ has exactly three values $c$ such that $f(c)=0$, its minimum is negative, its maximum is positive
$f$ has exactly three values $c$ such that $f(c)=0$, its minimum is positive, its maximum is negative
Originally, the interval is $[a,a]$, which is most likely a typo.