Let n 1 be an integer and suppose a0,a1, ,an are real numbers such that a0+a1++an =0 Show that the equation 2 n+1 a0 +a1x++anxn =0 has at least one...

Let n ≥ 1 be an integer and suppose a0,a1,··· ,an are real numbers such that a0+a1+···+an =0

Show that the equation

2 n+1

a0 +a1x+···+anxn =0

has at least one solution x0 with x0 ∈ (0, 1). Note:

You can assume without proof standard results on derivatives such as for example (xi)′ = ixi−1 for integers i ≥ 0.