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. \(Y\left( x \right) = \sqrt[3]{{x - 7}}\)
18. \(f\left( t \right) = {\left( {{t^3} - 25t} \right)^{\frac{2}{3}}}\)
19. \(h\left( x \right) = \sqrt[5]{x}\,\,{\left( {2x + 8} \right)^2}\)
20. \(Q\left( w \right) = \left( {6 - {w^2}} \right)\,\,\,\sqrt[3]{{{w^2} - 4}}\)
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.