Let $\mathbb { R } =$ the set of real numbers. A continuous function $f : \mathbb { R } \rightarrow \mathbb { R }$ satisfies $f ( 1 ) = 1$, $f ( 2 ) = 4 , f ( 3 ) = 9$ and $f ( 4 ) = 16$. Answer the independent questions below by choosing the correct option from the given ones. a) Which of the following values must be in the range of $f$? Options: 5 25 both neither Answer: $\_\_\_\_$ b) Suppose $f$ is differentiable. Then which of the following intervals must contain an $x$ such that $f ^ { \prime } ( x ) = 2 x$? Options: $( 1,2 )$ $( 2,4 )$ both neither Answer: $\_\_\_\_$ c) Suppose $f$ is twice differentiable. Which of the following intervals must contain an $x$ such that $f ^ { \prime \prime } ( x ) = 2$? Options: $(1,2)$ $(2,4)$ both neither Answer: $\_\_\_\_$ d) Suppose $f$ is a polynomial, then which of the following are possible values of its degree? Options: 3 4 both neither Answer: $\_\_\_\_$
Let $\mathbb { R } =$ the set of real numbers. A continuous function $f : \mathbb { R } \rightarrow \mathbb { R }$ satisfies $f ( 1 ) = 1$, $f ( 2 ) = 4 , f ( 3 ) = 9$ and $f ( 4 ) = 16$. Answer the independent questions below by choosing the correct option from the given ones.\\
a) Which of the following values must be in the range of $f$?
Options: 5\quad 25\quad both\quad neither
Answer: $\_\_\_\_$\\
b) Suppose $f$ is differentiable. Then which of the following intervals must contain an $x$ such that $f ^ { \prime } ( x ) = 2 x$?
Options: $( 1,2 )$\quad $( 2,4 )$\quad both\quad neither
Answer: $\_\_\_\_$\\
c) Suppose $f$ is twice differentiable. Which of the following intervals must contain an $x$ such that $f ^ { \prime \prime } ( x ) = 2$?
Options: $(1,2)$\quad $(2,4)$\quad both\quad neither
Answer: $\_\_\_\_$\\
d) Suppose $f$ is a polynomial, then which of the following are possible values of its degree?
Options: 3\quad 4\quad both\quad neither
Answer: $\_\_\_\_$