Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function (meaning $f'$ and $f''$ exist) such that $f(0)=f'(0)=0$, $f(1) = 1$, and $f'(1) = 4$.
Each of the following statement follows from either the Intermediate Value Theorem, the Extreme Value Theorem, or the Mean Value Theorem. Determine which one in each case.
$f''(x)=f'(x)+3$ for some $x$ in $[0,1]$ ---------EVT
$f$ and $f'$ are bounded on $[0,1]$--------------MVT
$f''(x) = 4$ for some $x$ in $[0,1]$.-------MVT
$f'(x) = 2$ for some $x$ in $[0,1]$.=----------------IVT
I am not centain about my answers, escpecially the first one, I am trying to understand these theorems and please let me know if I am correct! Thanks! I appreciate your help!