A polynomial function is defined by

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

Problem

A polynomial function

p

is defined by

p(x)=2x^{4}-3x^{3}+5x-7

. The graph of

p

is smooth for all real numbers. Answer the following about the differentiability of

p

x=1

. (a) State two different reasons, each stemming from a distinct formal definition or theorem, that guarantee the derivative

p'(1)

exists. (b) Describe, in words, the geometric meaning of the numerical value of

p'(1)

with respect to the graph of

y=p(x)

near the point

(1,p(1))