Question

Consider the function f(x)=x^3−2x+2 (a) Apply the Intermediate Value Theorem to prove that f(x) must have a root somewhere in the interval [−3,0]. (b) Start with an initial guess of x0 =−2 and apply the Newton-Raphson method to find an approximation to the root of f(x) in the interval [−3,0].

Answer 1

To apply the Intermediate Value Theorem, show that for any two values a and b in the interval [-3,0], f(a) and f(b) have opposite signs. To apply the Newton-Raphson method, start with an initial guess x0 = -2 and iterate until an accurate approximation is reached.

Step-by-step explanation:

To apply the Intermediate Value Theorem, we need to show that for any two values a and b in the interval [-3,0], f(a) and f(b) have opposite signs.

In this case, we have f(-3) = -13 and f(0) = 2.

Since -13 is negative and 2 is positive, we can conclude that there must be at least one root in the interval [-3,0].

To apply the Newton-Raphson method, we start with an initial guess x0 = -2.

Then, we iterate using the formula:

xn+1 = xn - f(xn)/f'(xn)

We continue this iteration until we reach an approximation that is accurate enough.