answer:To determine the critical values of the expression (frac{x^{2} - 2x + 2}{2x - 2}) within the range (-4 < x < 1), we need to follow these steps:1. Compute the first derivative of the expression.2. Find the values of (x) where the first derivative is zero or undefined (these are the critical points).3. Evaluate the expression at these critical points and within the given range to determine the nature of the critical values.Let's implement this step-by-step in Python using SymPy.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the expressionexpression = (x2 - 2*x + 2) / (2*x - 2)# Compute the first derivative of the expressionfirst_derivative = sp.diff(expression, x)# Solve for critical points where the first derivative is zero or undefinedcritical_points = sp.solve(first_derivative, x)# Filter critical points to ensure they are within the range -4 < x < 1valid_critical_points = [point for point in critical_points if point > -4 and point < 1]# Add the boundary points to the list of points to checkvalid_critical_points.extend([-4 + 1e-5, 1 - 1e-5]) # Slightly inside the boundary to avoid undefined behavior# Evaluate the expression at the critical points and boundary pointsvalues = [expression.subs(x, point).evalf() for point in valid_critical_points]min_value = min(values)max_value = max(values)(min_value, max_value)``````output(-50000.0000052275, -1.00000000000000)```The output indicates that within the range (-4 < x < 1), the critical values of the expression (frac{x^{2} - 2x + 2}{2x - 2}) include a minimum value of approximately (-50000) and a maximum value of (-1).To determine the correct answer choice, let's analyze these results:- The minimum value is (-50000), which is not one of the provided options.- The maximum value is (-1), which matches option (E).Thus, the critical values analysis within the given range leads us to conclude:(boxed{E}) Maximum value is -1.

answer:Let's solve this problem step-by-step, incorporating mathematical reasoning and using Python to verify our final result. Step-by-Step SolutionThe problem requires us to find non-empty subsets ( A ) of ({1, 2, 3, 4, 5, 6, 7}) such that for any ( a in A ), ( 8 - a ) is also in ( A ).1. Identify possible pairs: We need to identify all pairs ( (a, 8 - a) ) within the set ( {1, 2, 3, 4, 5, 6, 7} ). These pairs should cover all the elements in such a way that if one element is included in ( A ), the corresponding element by ( 8-a ) should also be in ( A ).2. Pairs within the set: Calculating the pairs that fit the condition: [ begin{align*} & (1, 7) & (2, 6) & (3, 5) & (4, 4) quad (text{self-pair}) end{align*} ]3. Subset Selection: For each possible pair, we have the choice of including that pair in ( A ) or not, except for the self-pair (4, 4), where the element 4 has to be included if selected.4. Counting Choices: - Each of the pairs ((1, 7)), ((2, 6)), ((3, 5)) gives us 2 choices (include both elements in ( A ) or exclude them from ( A )). - The self-pair ((4)) gives 2 choices (include 4 or not include 4).5. Calculate Total Number of Sets: Using the multiplication principle: [ 2 (text{choices for } {1,7}) times 2 (text{choices for } {2,6}) times 2 (text{choices for } {3,5}) times 2 (text{choices for } {4}) = 2^4 = 16 ]6. Exclude Empty Set: We need to exclude the empty set from our count, as the problem specifies non-empty subsets. [ text{Number of non-empty subsets} = 16 - 1 = 15 ] Verification by CodeWe'll use Python to generate and count these subsets methodically.```pythonfrom itertools import product # Define the pairs based on the problem statement pairs = [(1, 7), (2, 6), (3, 5)] # For each pair, we can either include or exclude, and we treat element 4 separately in its own category all_combinations = list(product([True, False], repeat=len(pairs))) # Variable to keep count of non-empty valid sets valid_subset_count = 0 # Iterate through each combination and calculate total number of valid subsets for combination in all_combinations: subset = set() for include, (a, b) in zip(combination, pairs): if include: subset.add(a) subset.add(b) # We also consider whether to include '4' or not for include_four in [True, False]: subset_with_four = subset.copy() if include_four: subset_with_four.add(4) if subset_with_four: valid_subset_count += 1 print(valid_subset_count)``````output15```Using both our step-by-step reasoning and the verification through the Python code, we find that the number of non-empty subsets ( A ) of ({1, 2, 3, 4, 5, 6, 7}) such that for any ( a in A ), ( 8 - a ) is also in ( A ) is:[boxed{15}]

answer:The binomial coefficient (binom{8}{0}) can be computed using the formula:[binom{n}{k} = frac{n!}{k!(n - k)!}]For (binom{8}{0}):[binom{8}{0} = frac{8!}{0!(8 - 0)!} = frac{8!}{0! cdot 8!}]Given that (0! = 1), the calculation simplifies to:[binom{8}{0} = frac{8!}{1 cdot 8!} = frac{8!}{8!} = 1]We can use Python code to verify this result. Let's calculate it using Python.```pythonimport math# Calculate binomial coefficientn = 8k = 0binomial_coefficient = math.comb(n, k)print(binomial_coefficient)``````output1```The binomial coefficient (binom{8}{0}) has been calculated to be (boxed{1}).

answer:Let's first understand the problem and then identify a method to solve it using Python.We need to find out how many integers between 2010 and 2099, when split between the second and third digits and multiplied, result in a perfect square. To do this:1. We'll iterate through each number in the range 2010 to 2099.2. For each number, we'll split it between the second and third digits to form two numbers.3. We'll multiply these two numbers.4. We'll check if the product is a perfect square.5. We'll count how many such numbers exist.Let's write Python code to achieve this.```pythonimport mathdef is_perfect_square(n): """Check if n is a perfect square.""" return int(math.isqrt(n))2 == ndef count_special_numbers(start, end): count = 0 for num in range(start, end + 1): # Split the number into two parts based on the problem's description str_num = str(num) part1 = int(str_num[:2]) # First two digits part2 = int(str_num[2:]) # Last two digits product = part1 * part2 if is_perfect_square(product): count += 1 return count# Define the rangestart = 2010end = 2099# Count the special numbers in the given rangespecial_numbers_count = count_special_numbers(start, end)print(special_numbers_count)``````output3```The total number of integers between 2010 and 2099 that have the property described is (boxed{3}).

answer:To solve the problem, we can use the following steps:1. Define the variables and given information: - Let ( v_B ) be the speed of boy ( B ) in miles per hour (mph). - Then the speed of boy ( A ) is ( v_B - 4 ) mph. - The distance between Port Jervis and Poughkeepsie is 60 miles. - Boy ( A ) meets boy ( B ) 12 miles from Poughkeepsie.2. Use the fact that when boy ( B ) reaches Poughkeepsie, he turns back and meets boy ( A ). - The time spent by ( B ) to travel 60 miles to Poughkeepsie is ( frac{60}{v_B} ) hours. - Let ( t ) be the time it takes from when boy ( B ) reaches Poughkeepsie to when they meet. - During this time ( t ), they together cover a distance of 48 miles (60 - 12 miles).3. The sum of their speeds times ( t ) should equal 48 miles: - ( (v_B + (v_B - 4)) cdot t = 48 ).4. We need to find the combined time ( t + frac{60}{v_B} ) and solve for the speed of boy ( A ).Let's solve this step-by-step using Python:```pythonfrom sympy import symbols, Eq, solve# Define the variablesv_B = symbols('v_B') # speed of B in mphv_A = v_B - 4 # speed of A in mph# Distance and the meeting point constraintdistance_total = 60 # total distance in milesdistance_meeting = 12 # distance from Poughkeepsie where they meet in milesdistance_remaining = distance_total - distance_meeting# Time taken by B to go from start to Poughkeepsietime_B_to_Poughkeepsie = distance_total / v_B# Time taken by B and A to meet after B reaches Poughkeepsiet = symbols('t')distance_cover_together = distance_total - distance_meeting# Equation derived from the distance covered together till they meet after B reaches Poughkeepsieeq = Eq((v_B + (v_B - 4)) * t, distance_cover_together)# Solve for tsolutions_t = solve(eq, t)time_t = solutions_t[0]# Now total time A and B travel when they meettime_A_meeting_total = time_t + time_B_to_Poughkeepsie# Using speed equation for A from start to meeting pointeq_A_speed = Eq((v_B - 4) * time_A_meeting_total, distance_total - distance_remaining)# Solve for speed of B and find speed of A (v_A)solutions_vB = solve(eq_A_speed, v_B)speed_of_B = solutions_vB[0]speed_of_A = speed_of_B - 4print(speed_of_A.evalf())``````output-2.52752523165195```It appears that the solve() method has returned a negative value for the speed of Boy (A), which is not physically possible. This suggests a possible error in our setup or an incorrect assumption. Let’s re-examine our assumptions and solve the problem step-by-step more methodically.We know the following:- ( v_B cdot t_B = 60 )- ( v_A cdot (t_B + t) = 60 - 12 = 48 )- ( v_A = v_B - 4 )- ( v_B cdot t = v_A cdot t + 12 )Firstly, let's reconsider our earlier equations and use them step-by-step to find an accurate solution.1. Time for (B) to reach Poughkeepsie is ( t_B = frac{60}{v_B} )2. Simultaneous time for (A) and (B) post (B)’s return ( t_A = t + t_B )3. Substitute ( t ) and ( t_B) in main speed equations.Let’s update our code with this approach:```python# Define the variablesv_B = symbols('v_B') # speed of B in mphv_A = v_B - 4 # speed of A in mph# Distance where A and B meet from startdistance_meeting_from_start = 60 - 12 # 48 miles from start# Time for B to travel 60 miles to Poughkeepsietime_B_to_Poughkeepsie = 60 / v_B# Time for A when they meettime_A_meeting = time_B_to_Poughkeepsie + (12 / v_B)# Distance travelled by A to meeting pointdistance_A_meeting = (v_B - 4) * time_A_meeting# Set up the equation for A's distance to meeting pointdistance_eq = Eq(distance_A_meeting, distance_meeting_from_start)# Solve for the speed of B (v_B)solutions_vB = solve(distance_eq, v_B)# Filter realistic positive solutionspeed_of_B = [speed for speed in solutions_vB if speed > 0][0]# Calculate speed of Aspeed_of_A = speed_of_B - 4print(speed_of_A.evalf())``````output8.00000000000000```The calculated speed of boy ( A ) is:[boxed{8 text{ mph}}]Hence, the rate of ( A ) was ( 8 text{ mph} ).

answer:To solve this problem, we need to find the value of (a_9) given the conditions. Let's analyze the given properties and how we can use SymPy to find the answer.1. Understand the Property: We are given that for any (p, q in mathbb{N}^*) (the set of all positive integers), ( a_{p+q} = a_p cdot a_q ).2. Given Value: We are given that (a_2 = 4).3. Determine the Form of (a_n): The property (a_{p+q} = a_p cdot a_q) is a well-known functional equation which suggests that (a_n) can be expressed in the form (a_n = a_1^n). This is because the only functions that satisfy this multiplicative property are of the form (a_n = c^n) for some constant (c).4. Find (a_1): Using the given value (a_2 = 4), we can find the constant (c) by solving (a_2 = c^2 = 4).5. Find (a_9): Once we have (c), we can compute (a_9 = c^9).Let's implement this using Python and SymPy:```pythonimport sympy as sp# Given informationa_2 = 4# Solve for the constant a_1 = cc = sp.symbols('c')equation = c2 - a_2solution = sp.solve(equation, c)# We need the positive value of cc_value = [sol for sol in solution if sol > 0][0]# Calculate a_9a_9 = c_value9print(a_9)``````output512```Based on our calculations using Python and SymPy, the value of (a_9) is (boxed{512}).