pls show work/steps clearly

Sketch the graph of the rational function 2x2 x2 1 f(X) = by determining the answers to the following parts. You must show calculations for each step and your graph must be derived from your work (meaning, do not graph using Desmos). (A) State the domain in interval notation. [1 mark]. (B) Determine any intercepts, written as points [2 marks]. (C) State the equations of any vertical asymptotes and use one-sided limits to determine the local behavior around the asymptotes. [5 marks]. (D) Use limits at infinity to calculate the equation of any horizontal asymptotes. [2 marks]. (E) Use the first derivative to determine the intervals where the function increases and decreases. [6 marks]. (F) Using the results from your sign chart in (E), list any local extrema as points. [1 mark]. (G) Using the second derivative, determine the intervals where the function is concave up and down. [6 marks]. (H) List any inflection points. [1 mark]. (I) Graph the function. Make sure you label everything you found in the steps above. [3 marks]. Exercise 1. Verify that the function f(x) = 2x2 - 8x + 6 defined over the interval [1, 3] satisfies the conditions of Rolle's theorem. Find all values of c guaranteed by Rolle's theorem. 2x+15 x+5 theorem on the interval [0, 20]. Find all values of c guaranteed by the MVT. Exercise 2. Verify that the function f (x) = satisfies the conditions of the Mean Value Calculate the antiderivative, F(x), of the function f(x) = 2x + 6 cos x, where F(T) = 12 + 2. Calculate the differential of y = x2 - 5x - 6, and evaluate it for x = 2 with dx = 0.1.Suppose you have 800 metres of fencing to make a pen for cattle. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area? [4 marks]. Please give a rough sketch of the problem to indicate your understanding. [1 mark]