Task $3.$

Given,

$f(x)=\frac{-1}{13}{x}^3+2x^2-9\mathrm{.}5x-10$

a$.($i$)$ Write a python function that takes an input function and a list of intervals as a list and returns a dictionary that contains either root exists

or not in each intervals. Determine if root exists in $(-20,-10),(-10,0),(0,10),(10,20),(20,30).$

$($ii$)$ Verify your method by ploting the function and the intervals.

$[]$ #$3$a$\_$i This cell should print

intervals $=[(-20,-10),(-10,0),(0,10),(10,20),(20,30)]$# This snippet will be given in the question

$[]$ #$3$a$\_$ii This cell should plot a graph.

b$.$ Using Bisection method, find roots of the function $3\mathrm{.}1$ in these intervals $(-20,-10),(-10,0),(0,10),(10,20),(20,30)$ where root

exists. The value of machine epsilon is

You can reuse the function of Task $3.$a to find out the intervals that contains root. Note: You should return $3$ different roots for the function $(3\mathrm{.}1).$

$[]$ #$3$b This cell should print

c$.$ Plot the $f(x)$ along with the roots to check if your method is working correctly.

$[]$ #$3$c This cell plot a graph.