Consider the polynomial

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

h(x)=4x^{5}-10x^{4}+15x^{3}-10x^{2}+5x

. Answer each part thoroughly, presenting general formulas where requested. (a) Derive an explicit expression for

h^{(n)}(x)

valid for every integer

n

with

0\le n\le5

. Write your result in closed form using factorial or falling-factorial notation. (b) Verify algebraically that

h^{(n)}(x)=0

for all integers

n\ge6

. (c) Using your formula from part (a), determine

h^{(n)}(0)

for each

n

with

0\le n\le5

. Present your answers in a clear table and describe any pattern you observe in these values.