Let f(x) = a₀+a₁x+a₂x², a₂ 0, be a quadratic function and g(x) = b₀+b₁x+b₂x²+b₃x³, b₃ 0, be a cubic polynomial.

1. Prove that f(x) will always be either concave up or concave down.

2. Prove that g(x) must have exactly one inflection point.