Question

The math assignment was due tomorrow so I need the answers with steps as soon as possible and I need it around today so please hurry

1. (15 marks) Find the domain of \(f(x)=2 x-\frac{1}{x^{2}}\) and find the intervals on which the function is increasing or decreasing.

2. (15 marks) For \(f(x)=\frac{x^{3}}{x+1}\), find the local maximum and minimum.

3. (15 marks) Find the absolute maximum and minimum of the function \(f(x)=\frac{x}{1+x^{2}}\) on the closed interval \([-1,2]\).

4. (20 marks) Sketch the graph of the curve \(f(x)=\frac{x^{3}}{x^{2}+1}\).

5. (20 marks) Consider the function

$$ g(x)=\frac{x^{2}-16}{x-5} $$

where \(x \neq 5\).

(a) Find all critical points of the function. Determine the intervals in which \(g(x)\) is increasing and the intervals in which \(g(x)\) is decreasing. Hence, or otherwise, find all the local (relative) maxima and local (relative) minima of the function.

(b) Find the intervals in which \(g(x)\) is concave up and the intervals in which \(g(x)\) is concave down. Hence determine the points of inflection of the function.

(c) Find all asymptotes of the function (including vertical, horizontal and inclined asymptotes). Sketch the graph of \(g(x)\).

6. (15 marks) Given that \(\int_{1}^{4} f(x) d x=5, \int_{3}^{4} f(x) d x=7\) and \(\int_{1}^{8} f(x) d x=11\), find the following:

(a) \(\int_{4}^{8} f(x) d x ;\) (b) \(\int_{4}^{3} f(x) d x ;\) (c) \(\int_{1}^{3} f(x) d x\).