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### Section 4.2 : Critical Points

For problems 1 - 43 determine the critical points of each of the following functions. Note that a couple of the problems involve equations that may not be easily solved by hand and as such may require some computational aids. These are marked are noted below.

1. $$R\left( x \right) = 8{x^3} - 18{x^2} - 240x + 2$$
2. $$f\left( z \right) = 2{z^4} - 16{z^3} + 20{z^2} - 7$$
3. $$\displaystyle g\left( z \right) = 8 - 12{z^5} - 25{z^6} + \frac{90}{7}{z^7}$$
4. $$g\left( t \right) = 3{t^4} - 20{t^3} - 132{t^2} + 672t - 4$$
Note : Depending upon your factoring skills this may require some computational aids.
5. $$\displaystyle h\left( x \right) = 10{x^2} - 15{x^3} + \frac{15}{2}{x^4} - {x^5}$$
Note : Depending upon your factoring skills this may require some computational aids.
6. $$P\left( w \right) = {w^3} - 4{w^2} - 7w - 1$$
7. $$A\left( t \right) = 7{t^3} - 3{t^2} + t - 15$$
8. $$a\left( t \right) = 4 - 2{t^2} - 6{t^3} - 3{t^4}$$
9. $$f\left( x \right) = 3{x^4} - 20{x^3} + 6{x^2} + 120x + 5$$
Note : Depending upon your factoring skills this may require some computational aids.
10. $$h\left( v \right) = {v^5} + {v^4} + 10{v^3} - 15$$
11. $$g\left( z \right) = {\left( {z - 3} \right)^5}{\left( {2z + 1} \right)^4}$$
12. $$R\left( q \right) = {\left( {q + 2} \right)^4}{\left( {{q^2} - 8} \right)^2}$$
13. $$f\left( t \right) = {\left( {t - 2} \right)^3}{\left( {{t^2} + 1} \right)^2}$$
14. $$\displaystyle f\left( w \right) = \frac{{{w^2} + 2w + 1}}{{3w - 5}}$$
15. $$\displaystyle h\left( t \right) = \frac{{3 - 4t}}{{{t^2} + 1}}$$
16. $$\displaystyle R\left( y \right) = \frac{{{y^2} - y}}{{{y^2} + 3y + 8}}$$
17. $$Y\left( x \right) = \sqrt{{x - 7}}$$
18. $$f\left( t \right) = {\left( {{t^3} - 25t} \right)^{\frac{2}{3}}}$$
19. $$h\left( x \right) = \sqrt{x}\,\,{\left( {2x + 8} \right)^2}$$
20. $$Q\left( w \right) = \left( {6 - {w^2}} \right)\,\,\,\sqrt{{{w^2} - 4}}$$
21. $$\displaystyle Q\left( t \right) = 7\sin \left( \frac{t}{4} \right) - 2$$
22. $$g\left( x \right) = 3\cos \left( {2x} \right) - 5x$$
23. $$f\left( x \right) = 7\cos \left( x \right) + 2x$$
24. $$h\left( t \right) = 6\sin \left( {2t} \right) + 12t$$
25. $$\displaystyle w\left( z \right) = {\cos ^3}\left( \frac{z}{5} \right)$$
26. $$U\left( z \right) = \tan \left( z \right) - 4z$$
27. $$h\left( x \right) = x\cos \left( x \right) - \sin \left( x \right)$$
28. $$h\left( x \right) = 2\cos \left( x \right) - \cos \left( {2x} \right)$$
29. $$f\left( w \right) = {\cos ^2}\left( w \right) - {\cos ^4}\left( w \right)$$
30. $$F\left( w \right) = {{\bf{e}}^{14w + 3}}$$
31. $$g\left( z \right) = {z^2}{{\bf{e}}^{1 - z}}$$
32. $$A\left( x \right) = \left( {3 - 2x} \right){{\bf{e}}^{{x^{\,2}}}}$$
33. $$P\left( t \right) = \left( {6t + 1} \right){{\bf{e}}^{8t - {t^{\,2}}}}$$
34. $$f\left( x \right) = {{\bf{e}}^{3 + {x^{\,2}}}} - {{\bf{e}}^{2{x^{\,2}} - 4}}$$
35. $$f\left( z \right) = {{\bf{e}}^{{z^{\,2}} - 4z}} + {{\bf{e}}^{8z - 2{z^2}}}$$
36. $$h\left( y \right) = {{\bf{e}}^{6{y^{\,3}} - 8{y^{\,2}}}}$$
37. $$g\left( t \right) = {{\bf{e}}^{2{t^{\,3}} + 4{t^{\,2}} - t}}$$
38. $$Z\left( t \right) = \ln \left( {{t^2} + t + 3} \right)$$
39. $$G\left( r \right) = r - \ln \left( {{r^2} + 1} \right)$$
40. $$A\left( z \right) = 2 - 6z + \ln \left( {8z + 1} \right)$$
41. $$f\left( x \right) = x - 4\ln \left( {{x^2} + x + 2} \right)$$
42. $$g\left( x \right) = \ln \left( {4x + 2} \right) - \ln \left( {x + 4} \right)$$
43. $$h\left( t \right) = \ln \left( {{t^2} - t + 1} \right) + \ln \left( {4 - t} \right)$$
44. The graph of some function, $$f\left( x \right)$$, is shown. Based on the graph, estimate the location of all the critical points of the function. 45. The graph of some function, $$f\left( x \right)$$, is shown. Based on the graph, estimate the location of all the critical points of the function. 46. The graph of some function, $$f\left( x \right)$$, is shown. Based on the graph, estimate the location of all the critical points of the function. 