Question 12 Prove that there is no p P3 such that px0forallx

Question 12. Prove that there is no p P3 such that p(x)=0forallxR, alsowithp(0)=1andp(1)=2. (Hint... try contradiction!) You are free to use Lemma 6.14 of the supplementary document here (look it up to see what it says!), and also, if you would like, there is a nice and different proof that invokes the mean value theorem for derivatives (remember it from calculus?!?!? if not, then look it up somewhere like a textbook or online).

Solution

Assume such a polynomial exists

p(x)=a+bx+cx^2+dx^3

p(0)=1 =a

p\'(x)=b+2cx+3dx^2=0 for all x

Setting x=0 gives b=0

p\'(x)=2cx+3dx^2

Setting x=1 gives

2c+3d=0

Setting x=-1 givs

-2c+3d=0

Hence, c=d=0

So, p(x)=1

But, p(1)=2. HEnce a contradiction

HEnce no such p exists