If f(x_n) * f(x_{n+1})

Question

If f(x_n) * f(x_{n+1}) < 0 then

A. There must be a root of f(x) between x_n and x_{n+1}
B. There need not be a root of f(x) between x_n and x_{n+1}
C. There is a fourth derivative of f(x) with respect to x that vanishes at x_1
D. The fourth derivative of f(x) with respect to x vanishes at x_{n+1}

Accepted Answer

A. There must be a root of f(x) between x_n and x_{n+1} If f(x_n) * f(x_{n+1}) < 0, the two function values have opposite signs. By the Intermediate Value Theorem (IVT), if f is continuous on [x_n, x_{n+1}], then there must exist at least one point c in (x_n, x_{n+1}) where f(c) = 0. This is the sign-change root-bracketing condition used in numerical methods like bisection.

If f(x_n) * f(x_{n+1}) < 0 then A. There must be a root of f(x) between x_n and x_{n+1} B. There need not be a root of f(x) between x_n and x_{n+1} C. There is a fourth derivative of f(x) with respect to x that vanishes at x_1 D. The fourth derivative of f(x) with respect to x vanishes at x_{n+1}