Mathematics

Problem

Consider the quartic polynomial

q(x)=x^{4}+px^{2}+qx+r

, where

p,q,

and

r

are real parameters. (a) Find

q''(x)

in its simplest form. (b) Derive the conditions on

p

and

q

that would make

q

concave up on

(-\infty,0)

and concave down on

(0,\infty)

simultaneously. Show all reasoning. (c) Prove that no real numbers

p

and

q

satisfy the simultaneous concavity requirements in part (b), regardless of the value of

r