1. Show that 3 x = 5x for some x ∈ (0, 1). 2. Show that the equation 5 x = x 4 has at least one real solution. 3. Show that any polynomial of odd degree has at least one real root. 4. Suppose that f :...

1. Show that 3^x
= 5x for some x ∈ (0, 1).

2. Show that the equation 5^x
= x⁴
has at least one real solution.

3. Show that any polynomial of odd degree has at least one real root.

4. Suppose that f : [a, b] → [a, b] is continuous. Prove that f has a fixed point. That is, prove that there exists c ∈ [a, b] such that f (c) = c.