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.
You are supposed to think about how $f$ might meet the requirements. You could start with the interval and think about min/max at the endpoints. An example for 1 is below. It is just $x^3-x$ on $[-\frac 32,\frac 32]$. Yes there are three cases where the function is zero, but they did not say exactly two. You can change the function to make it exactly two if you want. How would you do so?