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. $f\left( x \right) = 8{x^3} + 81{x^2} - 42x - 8$ on $\left[ { - 8,2} \right]$ Solution
2. $f\left( x \right) = 8{x^3} + 81{x^2} - 42x - 8$ on $\left[ { - 4,2} \right]$ Solution
3. $R\left( t \right) = 1 + 80{t^3} + 5{t^4} - 2{t^5}$ on $\left[ { - 4.5,\,\,4} \right]$ Solution
4. $R\left( t \right) = 1 + 80{t^3} + 5{t^4} - 2{t^5}$ on $\left[ {0,7} \right]$ Solution
5. $h\left( z \right) = 4{z^3} - 3{z^2} + 9z + 12$ on $\left[ { - 2,1} \right]$ Solution
6. $g\left( x \right) = 3{x^4} - 26{x^3} + 60{x^2} - 11$ on $\left[ {1,5} \right]$ Solution
7. $Q\left( x \right) = {\left( {2 - 8x} \right)^4}{\left( {{x^2} - 9} \right)^3}$ on $\left[ { - 3,3} \right]$ Solution
8. $h\left( w \right) = 2{w^3}{\left( {w + 2} \right)^5}$ on $\left[ { - {\displaystyle \frac{5}{2}},{\displaystyle \frac{1}{2}}} \right]$ Solution
9. $\displaystyle f\left( z \right) = \frac{{z + 4}}{{2{z^2} + z + 8}}$ on $\left[ { - 10,0} \right]$ Solution
10. $A\left( t \right) = {t^2}\,{\left( {10 - t} \right)^{\frac{2}{3}}}$ on $\left[ {2,\,\,10.5} \right]$ Solution
11. $f\left( y \right) = \sin \left( {{\displaystyle \frac{y}{3}}} \right) + {\displaystyle \frac{2y}{9}}$ on $\left[ { - 10,15} \right]$ Solution
12. $g\left( w \right) = {{\bf{e}}^{{w^{\,3}} - 2{w^{\,2}} - 7w}}$ on $\left[ { - {\displaystyle \frac{1}{2}},\,\,{\displaystyle \frac{5}{2}}} \right]$ Solution
13. $R\left( x \right) = \ln \left( {{x^2} + 4x + 14} \right)$ on $\left[ { - 4,2} \right]$ Solution