1. Intermediate Value Theorem Vince Varju

2. Definition The Intermediate Value Theorem states that if a function f is a continuous function on [a,b] then there exists a number c in (a,b) so that f(c) = n n - any number between f(a) and f(b) where f(a) ≠ f(b) IVT is used to find solutions or zeros in polynomials along a given Domain by testing points and determining if the the sign changes between those two points. f = x^2-2 test if there is a zero between (0,4) f(0) = -2 f(4) = 14 since there is a sign change, by the definition of IVT, there is a zero between 0 and 4.

3. Examples 1.f(x) = x^3 - 2x^2 + 15x - 10 determine if there is a zero between (0,2) f(0) = (0)^3 - 2(0)^2 + 15(0) - 10 = -10 f(2) = (2)^3-2(2)^2+15(2)-10 = 8-8+30-10 = 20 Because the sign changes along the Domain of (0,2), by the definition of IVT, there is a zero between 0 and 2. 2. f(x) = x^4-x+2 determine if there is a zero between (3,5) f(3) = (3)^4-3+2 = 81-1 = 80 f(5) = (5)^4-5+2 = 625-3 = 622 Because there is no sign change along the Domain of (3,5), there is no zero between 3 and 5.

4. Practice Problems 1. f(x) = x^5 - 5x^4 + 2x^3 - 6x^2 + 15 determine if there is a zero between 0 and 4 2. f(x) = x^3 - 7x^2 + 10x - 4 determine if there is a zero between 1 and 5 3. f(x) = -5x^2 + 8x + 2 determine if there is a zero between 0 and 1

5. Solutions 1. f(0) = 15 f(4) = -209 by the definition of IVT, there is a zero between 0 and 4. 2. f(1) = 0 f(5) = -4 by the definition of IVT, there is a zero between 1 and 5. 3. f(0) = 2 f(2) = 5 by the definition of IVT, there is not a solution between 0 and 1.