The first part is the intermediate value theorem.
The second bit is newton Rhapson method.
Newton's Method -- from Wolfram MathWorld
show that the cubic y=x^3+2x^2+5 has zero somewhere between x=-2.5 and x=-3. pick a suitable starting estimate of the root, x0, and perfrom 3 iterations of newtons method. test your final estimate to see how good it is.
how do i go about this?
(I can't work this out in my head!)
f(x) is continous since it's a polynomial (and all polynomials are continuous).
By the Intermediate Value Theorem f(x) must assume every value between f(-3) and f(-2.5) since it's continuous. Therefore . is a solution to f(x).