Question 69
Suppose $a$, $b$, and $c$ are real numbers, with $a \neq b$. Prove that the minimum value of the quadratic polynomial
$$
p(x) = 3x^2 + 2c x - (c(a+b) + (a^2 + ab + b^2)), \quad -\infty \lt x \lt \infty,
$$
is negative. Prove also that $p(a) + p(b)$ is positive.
$$
p(x) = 3x^2 + 2c x - (c(a+b) + (a^2 + ab + b^2)), \quad -\infty \lt x \lt \infty,
$$
is negative. Prove also that $p(a) + p(b)$ is positive.