Application of Rolle'sSuppose q is a nonzero function of a real-variable such thatu2q&nbsp;″(u)+uq′(u)=u2q(u)+q(u)&nbsp;for all u.

Question

Application of Rolle&#039;sSuppose q is a nonzero function of a real-variable such thatu2q&amp;nbsp;″(u)+uq′(u)=u2q(u)+q(u)&amp;nbsp;for all u.

Veronica Riddle · Accepted Answer

Step 1We assume that for all u, q is continuous, second differentiable and not uniformly zero on any interval.Assume that there exist x, y such that q(x)=q(y)=0.Since q is continuous, second differentiable and not uniformly zero on (x,y) it is either the case that there is a1. local maximum or a2. local minimum point (z,q(z)) of q on the open interval (x,y).Step 2In either case, q′(z)=0, so it follows thatz2q″(z)=(z2+1)q(z)So both q(z) and q&#039;&#039;(z) must have the same sign.But if (z,q(z)) is a global maximum q(z)&amp;gt;0 and qz)&amp;lt;0 and it it is a global minimum then q(z)&amp;lt;0 and qz)&amp;gt;0.This is a contradiction.So q cannot have distinct zeros unless q is uniformly zero on some interval.

Application of Rolle's Suppose q is a nonzero function

Answered question

Answer & Explanation

New Questions in Calculus 2