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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 $$\left[ {1,3} \right]$$