Let be an integer

MATHEMATICS · Subjective · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let n≥3n\ge 3 be an integer. (a) For the polynomial pn(x)=xnp_n(x)=x^{n}, prove that x=0x=0 is a critical point and that pn′′(0)=0p_n''(0)=0. (b) Show that if nn is odd, x=0x=0 is neither a point of local maximum nor a point of local minimum for pn(x)p_n(x). Give a rigorous argument based on the behaviour of pn′(x)p_n'(x) in a neighbourhood of 00. (c) Show that if nn is even and n≥4n\ge 4, then: (i) pn(n)(0)=n!≠0p_n^{(n)}(0)=n!\ne 0 while all derivatives of orders 3,4,…,n−13,4,\ldots,n-1 vanish at x=0x=0; (ii) x=0x=0 is a point of local minimum for pn(x)p_n(x). Justify each step clearly. (d) Summarise your results by stating how the parity of nn affects the nature of the critical point at the origin for the family pn(x)=xnp_n(x)=x^{n}.

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