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- Let f(x)=2x^2−5x+2 / 3x^2+11x+6This function has:1) A y-intercept at the point =2) x-intercepts at the point(s) =3) Vertical asymptotes at x = 4) Horizontal asymptote at y =Find a rational function f that satisfies all the given properties vertical asymptote at x = 4 and horizontal asymptote at y = 0Use limits to determine the equations for all horizontal asymptote. y = (1 - x2)/( x2 + 1)
- Let f(x) = 8 x2 − 4x . Then its first derivative is f'(x) = 32 − 16x (x2 − 4x)2 . a. Find the equation of the horizontal asymptote of the graph of f. (If there is none, enter "DNE".)x = b. Find the locations (x-values) of all vertical asymptotes of the graph of f. If there are more than one, enter them as a comma-separated list. If there are none, enter "DNE". c. Find the critical number(s) of f. If there are more than one, enter them as a comma-separated list. If there are none, enter "DNE".Let f(x) = (2x+1)/(x+1)2. (d) Determine the horizontal asymptote(s) of the curve y=f(x) (if any). (e) Find the intervals of increase/decrease of f, given that f'(x)=(-2x)/(x+1)3. (f) Determine the points at which f has a local maximum/minimum (if any).In the function: f(x)= (3x^2)ln(x) , x>0 What are the vertical asymptotes?
- What values of a and b make ƒ(x) = x3 + ax2 + bx have a. a local maximum at x = -1 and a local minimum at x = 3? b. a local minimum at x = 4 and a point of inflection at x = 1?sketch the graph of y = x^3 / (x^2 - 4), keep in mind the information provided:1. x and y intercept: (0, 0)2. Vertical asymptotes: x = 2, x = -23. There is no horizontal asymptote.4. Discontinuity at x = -2 and x = 2.5. First derivative: y’ = (x^4 - 12x^2) / (x^2 - 4)^26. Critical points: x = 0, x = 2√3, x = -2√37. Intervals of increase: (-∞, -2√3) U (2√3, ∞)8. Intervals of decrease: (-2√3, -2) U (-2, 0)9. Local maxima: (-3.46, -5.20) and (3.46, 5.20)10. Second derivative: y’’ = (8x^3 + 96x) / (x^2 - 4)^311. Critical numbers: (0, 0), (√12, 3√3), (-√12, -3√3)12. Point of inflection: (0, 0)13. Concave upward: (-2, 0) U (2, ∞)14. Concave downward: (-∞, -2) U (0, 2) A sketch of the graph (by hand) - Labeling of vertical asymptotes on the sketch - Labeling of horizontal asymptotes on the sketch - Labeling of x and y intercepts on the sketch - Labeling of maximum and minimum points on the sketch - Labeling of points of inflection on the sketch A sketch of the graph on Desmos with all key…Suppose that f(x) is a rational function with discontinuities at x = ±√2 whose first derivative is f'(x)= 4x/(x^2 -2)^2. Further, assume the function f(x) has vertical asymptotes x = ±√2 and horizontal asymptote y= 1.a. Determine all critical points of f. b. Find the intervals on which the graph of f is increasing or decreasing c. For each critical value determine if f(x) has a local maximum or local minimum
