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Section 4.4 : Finding Absolute Extrema

For each of the following problems determine the absolute extrema of the given function on the specified interval.

1. \(f\left( z \right) = 2{z^4} - 16{z^3} + 20{z^2} - 7\) on \(\left[ { - 2,6} \right]\)
2. \(f\left( z \right) = 2{z^4} - 16{z^3} + 20{z^2} - 7\) on \(\left[ { - 2,4} \right]\)
3. \(f\left( z \right) = 2{z^4} - 16{z^3} + 20{z^2} - 7\) on \(\left[ {0,2} \right]\)
4. \(Q\left( w \right) = 20 + 280{w^3} + 75{w^4} - 12{w^5}\) on \(\left[ { - 3,2} \right]\)
5. \(Q\left( w \right) = 20 + 280{w^3} + 75{w^4} - 12{w^5}\) on \(\left[ { - 1,8} \right]\)
6. \(\displaystyle g\left( z \right) = 8 - 12{z^5} - 25{z^6} + \frac{90}{7}{z^7}\) on \(\left[ { - 1,1} \right]\)
7. \(g\left( t \right) = 3{t^4} - 20{t^3} - 132{t^2} + 672t - 4\) on \(\left[ { - 5,8} \right]\)
   Note : Depending upon your factoring skills this may require some computational aids.
8. \(g\left( t \right) = 3{t^4} - 20{t^3} - 132{t^2} + 672t - 4\) on \(\left[ { - 2,8} \right]\)
   Note : Depending upon your factoring skills this may require some computational aids.
9. \(V\left( x \right) = 14{x^3} + 11{x^2} - 4x + 3\) on \(\left[ { - 1,1} \right]\)
10. \(a\left( t \right) = 4 - 2{t^2} - 6{t^3} - 3{t^4}\) on \(\left[ { - 2,1} \right]\)
11. \(h\left( x \right) = 8 + 3x + 7{x^2} - {x^3}\) on \(\left[ { - 1,5} \right]\)
12. \(f\left( x \right) = 3{x^4} - 20{x^3} + 6{x^2} + 120x + 5\) on \(\left[ { - 1,5} \right]\)
    Note : This problem will require some computational aids.
13. \(h\left( v \right) = {v^5} + {v^4} + 10{v^3} - 15\) on \(\left[ { - 3,2} \right]\)
14. \(g\left( z \right) = {\left( {z - 3} \right)^5}{\left( {2z + 1} \right)^4}\) on \(\left[ { - 1,3} \right]\)
15. \(R\left( q \right) = {\left( {q + 2} \right)^4}{\left( {{q^2} - 8} \right)^2}\) on \(\left[ { - 4,1} \right]\)
16. \(\displaystyle h\left( t \right) = \frac{{3 - 4t}}{{{t^2} + 1}}\) on \(\left[ { - 2,4} \right]\)
17. \(\displaystyle g\left( x \right) = \frac{{6 + 9x + {x^2}}}{{1 + x + {x^2}}}\) on \(\left[ { - 6,0} \right]\)
18. \(f\left( t \right) = {\left( {{t^3} - 25t} \right)^{\frac{2}{3}}}\) on \(\left[ {2,6} \right]\)
19. \(F\left( t \right) = 2 + {t^{\frac{2}{5}}}\,\left( {1 + t + {t^2}} \right)\) on \(\left[ { - 2,1} \right]\)
20. \(Q\left( w \right) = \left( {6 - {w^2}} \right)\,\,\,\sqrt[3]{{{w^2} - 4}}\) on \(\displaystyle \left[ { - 5,{\frac{1}{2}}} \right]\)
21. \(g\left( x \right) = 3\cos \left( {2x} \right) - 5x\) on \(\left[ {0,6} \right]\)
22. \(\displaystyle s\left( w \right) = 3w - 10\sin \left( {{\frac{w}{3}}} \right)\) on \(\left[ {10,38} \right]\)
23. \(f\left( x \right) = 7\cos \left( x \right) + 2x\) on \(\left[ { - 5,4} \right]\)
24. \(h\left( x \right) = x\cos \left( x \right) - \sin \left( x \right)\) on \(\left[ { - 15, - 5} \right]\)
25. \(g\left( z \right) = {z^2}{{\bf{e}}^{1 - z}}\) on \(\left[ {\displaystyle - {\frac{1}{2}},{\displaystyle \frac{5}{2}}} \right]\)
26. \(P\left( t \right) = \left( {6t + 1} \right){{\bf{e}}^{8t - {t^{\,2}}}}\) on \(\left[ { - 1,3} \right]\)
27. \(f\left( x \right) = {{\bf{e}}^{5 + 9x}} + {{\bf{e}}^{1 - 3x}} + 6\) on \(\left[ { - 1,0} \right]\)
28. \(h\left( y \right) = {{\bf{e}}^{6{y^{\,3}} - 8{y^{\,2}}}}\) on \(\left[ { - {\displaystyle \frac{1}{2}},1} \right]\)
29. \(Z\left( t \right) = \ln \left( {{t^2} + t + 3} \right)\) on \(\left[ { - 2,2} \right]\)
30. \(f\left( x \right) = x - 4\ln \left( {{x^2} + x + 2} \right)\) on \(\left[ { - 1,9} \right]\)
31. \(h\left( t \right) = \ln \left( {{t^2} - t + 1} \right) + \ln \left( {4 - t} \right)\) on \(\left[ {1,3} \right]\)