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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. \(Y\left( x \right) = \sqrt[3]{{x - 7}}\)
  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.
    This graph has no y scale.  It starts at approximately x=-4.8 in the 2nd quadrant and decreases until it reaches a valley at x=-3 and then increases until it reaches a peak at x=1.  It then decreases to a valley at x=4 and increases until it reaches a point in the 1st quadrant at approximately x=5.8.
  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.
    This graph has no y scale.  It starts at x=-5 in the 2nd quadrant and decreases until it reaches a valley at x=-4 and then increases going through a point at x=-1 perfectly flat and continuing to increases until it a peak at x=3.  It then decreases until it stops at z=4 in the 1st quadrant.
  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.
    This graph has no y scale.  It starts at x=-3 in the 2nd quadrant and decreases going through a point at x=-2 perfectly flat and continuing to decrease until it a valley at x=1.  It then increases and goes through a point at x=5 perfectly flat and continues to increases until it stops at z=6 in the 1st quadrant.