Does there exists a continuous function $f(x)$ such that $f(0)=10,f(2)=2, f'(x) \le 1,\forall x\in (0,2)?$

Question

Does there exists a continuous function $f(x)$ such that $f(0)=10,f(2)=2, f'(x) \le 1,\forall x\in (0,2)?$(Hint:-Use Mean Value theorem)

My Attempt Assume there exists a function satisfy Mean value theorem premises. So, there exists $c\in(0,2):$ $f(2)=f(0)=f'(c)(2-0)\leq 2$. We know that $f(2)-f(0)=2-10=-8$. which satisfies Mean value theorem. Does it imply the existence? Please help me.

Answer 1

$f(2) < f(0)$, so just a straight line connecting those points would have a negative slope.

You can verify its existence with MVT, of course.