Mathematics

Problem

Let

p(x)=x^{2}+1

and

q(x)=x^{3}-x

. Define

P(x)=p(x)\,q(x)

and

F(x)=P(x)+r(x)

where

r(x)=x^{5}+kx

and

k

is a real constant. a. (i) Prove that the product of an even function and an odd function is an odd function. b. (ii) Using the result of part (i), classify

P(x)

as even, odd, or neither, showing all necessary steps. c. (iii) Determine the value(s) of

k

for which the function

F(x)

becomes an even function. Show your complete working.

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