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13 Nov 2019

# Let F be differentiable on R, one version of Rolle's theorem states, "between the zeros of F there is a zero of F'." a) Verify this theorem for the polynomials Fi = (z-1)(z-2) F2 = (z-1)(x-2)(x-3), by finding the zeros of both F and F b) Now suppose that F is differentiable on R. Write down a one line statement in which Rolle's theorem is applied to F, and verify your new theorem for F2 c) Let G = xn + ax+ b, where n is a positive integer and a and b are non-zero real constants. Make use of part (a) to prove that: (i) if n is even then G has at most two zeros, (ii) if n is odd then G has at most three zeros.

Let F be differentiable on R, one version of Rolle's theorem states, "between the zeros of F there is a zero of F'." a) Verify this theorem for the polynomials Fi = (z-1)(z-2) F2 = (z-1)(x-2)(x-3), by finding the zeros of both F and F b) Now suppose that F is differentiable on R. Write down a one line statement in which Rolle's theorem is applied to F, and verify your new theorem for F2 c) Let G = xn + ax+ b, where n is a positive integer and a and b are non-zero real constants. Make use of part (a) to prove that: (i) if n is even then G has at most two zeros, (ii) if n is odd then G has at most three zeros.

Jamar FerryLv2

13 Nov 2019