\(\displaystyle f(x) \ = \ (3-\frac{1}{x})(x+5)\)
\(\displaystyle D_x\bigg[(3-\frac{1}{x})(x+5)\bigg] \ = \ (3-\frac{1}{x})(1)+(x+5)(\frac{1}{x^{2}})\)
\(\displaystyle = \ 3-\frac{1}{x}+\frac{1}{x}+\frac{5}{x^{2}} \ = \ 3+\frac{5}{x^{2}}\)
\(\displaystyle f(x) \ = \ (3-\frac{1}{x})(x=5) \ = \ (\frac{3x-1}{x})(x+5) \ = \ \frac{3x^{2}+14x-5}{x}\)
\(\displaystyle f \ ' \ (x) \ = \ \frac{x(6x+14)-(3x^{2}+14x-5)(1)}{x^{2}} \ = \ \frac{6x^{2}+14x-3x^{2}-14x+5}{x^{2}} \ = \ \frac{3x^{2}+5}{x^{2}} \ = \ 3+\frac{5}{x^{2}}\)
\(\displaystyle Note: \ No \ critical \ points, \ but \ two \ asymptotes, \ what \ are \ they? \ See \ graph.\)
[attachment=1:1imj7unc]abc.jpg[/attachment:1imj7unc]
\(\displaystyle Oops, \ I \ misread \ the \ function.\\)
\(\displaystyle f(x) \ = \ \frac{(3-\frac{1}{x})}{(x+5)} \ = \ \frac{3x-1}{x^{2}+5x}\)
\(\displaystyle f \ ' \ (x) \ = \ \frac{(x^{2}+5x)(3)-(3x-1)(2x+5)}{(x^{2}+5x)^{2}} \ = \ \frac{3x^{2}+15x-6x^{2}-13x+5}{(x^{2}+5x)^{2}}\)
\(\displaystyle = \ \frac{-3x^{2}+2x+5}{(x^{2}+5x)^{2}} \ = \ -\frac{3x^{2}-2x-5}{x^{2}(x+5)^{2}}\)
\(\displaystyle Two \ critical \ points, f \ ' \ (-1) \ = \ 0, \ f \ ' \ (5/3) \ = \ 0, \ see \ graph.\)
[attachment=0:1imj7unc]zzz.jpg[/attachment:1imj7unc]
