Let $f$ be a continuous map then which of the following is correct

Question

Let $f$ be a continuous function from $[0,4]$ to $[3,6]$ Then

(a) There must be a $x$ such that $f(x) = 4$

(b) There must be a $x$ such that $3f(x) = 2x+6$

(c)There must be a $x$ such that $2f(x) = 2x+6$

(d)There must be a $x$ such that $f(x) = x$

Using Intermediate Value theorem I know that (a) must be correct , but I have no idea for rest of the options any hints will be helpful.

Thank you.

Answer 1

Visual aid as hint - can you draw a graph in the shaded region starting from $x=0$ ending at $x=4$ that doesn't intersect (a) the blue line? (b) the green line? (c) yellow line? (d) black line?

Answer 2

The correct answer is (c).

Let $g(x)=2f(x)-2x-6$.

Then $g(0)=2f(0)-6\ge2(3)-6=0$ and $g(4)=2f(4)-14\le 2(6)-14<0$.

By IVT, there must be a $x$ such that $g(x)=0$.

For other options, we can find a $f(x)$ to avoid equality. (See Calvin Khor's answer)