Finding extrema in the interval

Question

Assignment:

Q1. Find all the critical numbers: f(x) = x - 1 / x + 3  .

Q2. Find all the extrema in the interval [0,2π ] for y = sinx + cosx .

Q3. Find the absolute maximum and the absolute minimum on the interval (1,4]. f(x) = x³ - 7x² + 12x -6 / x - 1

Q4. State why Rolle’s Theorem does not apply to the function f(x) = 2 / (x + 1)²  on the interval [-2,0].

Q5. Find all relative extrema of y = (x + 1)² (x - 2)  , include the designation of maximum or minimum.

Q6. Find all the points of inflection of the graph of the function f(x) = x⁴ -x³ .

Q7. Find the horizontal asymptote(s) for f(x) = 5x /√x² + 3  .

Q8. Find the limit_x→∞ a - bx⁴ /cx⁴ +x² .

Q9. Find two numbers whose product is maximum if the sum of the first number and five times the second is 80.

Q10. Calculate 3 iterations of Newton’s Method to approximate the real zero of f(x) = x³ + x +1.

Provide complete and step by step solution for the question and show calculations and use formulas.