Consider the function f:[12,1]→R defined by f(x)=42√x3−32√x−1.Consider the statements(I) The curve y=f(x) intersects the x-axis exactly at one point(II) The curve y=f(x) intersects the x-axis at x=cosπ12ThenOnly (II) is correctBoth (I) and (II) are incorrectOnly (I) is correctBoth (I) and (II) are correct

1. Suppose that, for x ≥ 0, the curve C has equation y = 3x − x√x.(a) Find the coordinates of the x-intercepts of C.(4 marks)(b) Find the coordinates of the stationary point of C and determine whether it isa maximum, a minimum or a point of inflection.(7 marks)(c) Sketch the curve C.

The graph of a function f is given in the figure.A curve is shown on the x y coordinate plane. It begins at the point (−2, −1), goes up and to the right, passes through the approximate point (−1, −0.2), and passes through the negative x-axis at the approximate point (−0.8, 0). It then continues up and right, passes through the positive y-axis at the point (0, 1), and reaches a high point at (1, 3). It then goes down and right, passes through the points (2, 2) and (3, 1), and ends at the approximate point (4, 0.5).(a)Find the value of f(1).(b)Estimate the value of f(−1).(c)For what values of x is f(x) = 1? (Enter your answers as a comma-separated list.) (d)Estimate the value of x such that f(x) = 0.x = (e)State the domain and range of f. (Enter your answers in interval notation.)domain range (f)On what interval is f increasing? (Enter your answer using interval notation.)

Suppose that, for x ≥ 0, the curve C has equation y = 3x − x√x.(a) Find the coordinates of the x-intercepts of C

y= f(x)=x^2-x-1 y = = f(x)=x^3-x^2-2x+1 Implement Bisection Method to find all the possible roots of these 2 given functions and verify it with built-in functions scipy.optimize.root() in the SciPy library. follow this given template # Root Finding Method import math import numpy as np import scipy as sp import matplotlib.pyplot as plt def plot_function(func, a, b): """ This function plot the graph of the input func within the given interval [a,b). """ # Your code goes here def bisection_method(func, a, b, tol=1e-6, max_iter=100): """ Bisection method to find the root of a function within a given interval. Parameters: - func: The function for which the root is to be found. - a, b: Interval [a, b] within which the root is searched for. - tol: Tolerance level for checking convergence of the method. - max_iter: Maximum number of iterations. Returns: - root: Approximation of the root. Example -------- >>> fun = lambda x: x**2 - x - 1 >>> root = bisection_method(fun, 1, 2, max_iter=20) """ # Check if the interval is valid (signs of f(a) and f(b) are different) # Your code goes here # Main loop starts here iter_count = 1 while iter_count <= max_iter: # your code goes here iter_count += 1 print("Warning! Exceeded the maximum number of iterations.") return root # Example usage: if __name__ == "__main__": # Define the function for which the root is to be found func = lambda x: x**2 - x - 1 # First Function # Uncomment the below line to use the Second Function # func = lambda x: x**3 - x**2 - 2*x + 1 # Second Function # Call plot_function to plot graph of the function # Your code goes here # Set the interval [a, b] for the search a_1 = 0; b_1 = 0; # For first root (change the values as required) a_2 = 0; b_2 = 0; # For second root (change the values as required) # Call the bisection method our_root_1 = # your code goes here our_root_2 = # your code goes here # Call SciPy method root, which we consider as a reference method. x0 = (a_1 + b_1)/2 sp_result_1 = sp.optimize.root(func, x0) sp_root_1 = sp_result_1.x.item() x0 = (a_2 + b_2)/2 sp_result_2 = sp.optimize.root(func, x0) sp_root_2 = sp_result_2.x.item() # Print the result print("1st root found by Bisection Method = {:0.8f}.".format(our_root_1)) print("1st root found by SciPy = {:0.8f}".format(sp_root_1)) print("2nd root found by Bisection Method = {:0.8f}.".format(our_root_2)) print("2nd root found by SciPy = {:0.8f}".format(sp_root_2))

"Quadratic functions have Answer 1 Question 16, or x𝑥-intercepts, in any other case where the function crosses the x𝑥-axis at two separate points.