MathDB

Two polynomials

P(x)=x^4+ax^3+bx^2+cx+d

and

Q(x)=x^2+px+q

have real coefficients, and

I

is an interval on the real line of length greater than

2

. Suppose

P(x)

and

Q(x)

take negative values on

I

, and they take non-negative values outside

I

. Prove that there exists a real number

x_0

such that

P(x_0)<Q(x_0)

2