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