b. Estimate a solution to the equation in the given interval using a root finder.


x^5+7x+5=0; (−1,0)

Verified step by step guidance

1. First, understand that the problem requires finding a root of the equation \(x^5 + 7x + 5 = 0\) within the interval \((-1, 0)\). A root finder method, such as the bisection method, can be used for this purpose.
2. Begin by evaluating the function at the endpoints of the interval. Calculate \(f(-1)\) and \(f(0)\) to check if there is a sign change, which indicates the presence of a root in the interval.
3. If there is a sign change, apply the bisection method: calculate the midpoint of the interval, \(c = \frac{-1 + 0}{2} = -0.5\), and evaluate \(f(c)\).
4. Determine which subinterval, \((-1, -0.5)\) or \((-0.5, 0)\), contains a sign change by checking the sign of \(f(c)\) and comparing it with the signs of \(f(-1)\) and \(f(0)\).
5. Repeat the bisection process on the subinterval with the sign change, continually halving the interval and evaluating the function at the new midpoints, until the interval is sufficiently small to estimate the root accurately.