Theorem: Let $f$ be continuous on $[a,\phantom{\rule{thinmathspace}{0ex}}b]$ and assume $f(a)\ne f(b)$. Then for every $\lambda $ such that $f(a)<\lambda <f(b)$, there exists a $c\in (a,\phantom{\rule{thinmathspace}{0ex}}b)$ such that $f(c)=\lambda $.

Question:

Suppose that $f:[0,1]\to [0,2]$ is continuous. Use the Intermediate Value Theorem to prove that their exists $c\in [0,1]$ such that:

$f(c)=2{c}^{2}$

Attempt:

I know that when we have the condition were $f:[a,b]\to [a,b]$, the method to prove that c exits, is the same method you would use to prove the fixed point theorem.

Unfortunately I don't have an example in my notes when we have $f:[a,b]\to [a,y]$. How would I use the IVT to answer the original question?

Question:

Suppose that $f:[0,1]\to [0,2]$ is continuous. Use the Intermediate Value Theorem to prove that their exists $c\in [0,1]$ such that:

$f(c)=2{c}^{2}$

Attempt:

I know that when we have the condition were $f:[a,b]\to [a,b]$, the method to prove that c exits, is the same method you would use to prove the fixed point theorem.

Unfortunately I don't have an example in my notes when we have $f:[a,b]\to [a,y]$. How would I use the IVT to answer the original question?