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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
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.
- \(R\left( x \right) = 8{x^3} - 18{x^2} - 240x + 2\)
- \(f\left( z \right) = 2{z^4} - 16{z^3} + 20{z^2} - 7\)
- \(\displaystyle g\left( z \right) = 8 - 12{z^5} - 25{z^6} + \frac{90}{7}{z^7}\)
- \(g\left( t \right) = 3{t^4} - 20{t^3} - 132{t^2} + 672t - 4\)
Note : Depending upon your factoring skills this may require some computational aids. - \(\displaystyle h\left( x \right) = 10{x^2} - 15{x^3} + \frac{15}{2}{x^4} - {x^5}\)
Note : Depending upon your factoring skills this may require some computational aids. - \(P\left( w \right) = {w^3} - 4{w^2} - 7w - 1\)
- \(A\left( t \right) = 7{t^3} - 3{t^2} + t - 15\)
- \(a\left( t \right) = 4 - 2{t^2} - 6{t^3} - 3{t^4}\)
- \(f\left( x \right) = 3{x^4} - 20{x^3} + 6{x^2} + 120x + 5\)
Note : Depending upon your factoring skills this may require some computational aids. - \(h\left( v \right) = {v^5} + {v^4} + 10{v^3} - 15\)
- \(g\left( z \right) = {\left( {z - 3} \right)^5}{\left( {2z + 1} \right)^4}\)
- \(R\left( q \right) = {\left( {q + 2} \right)^4}{\left( {{q^2} - 8} \right)^2}\)
- \(f\left( t \right) = {\left( {t - 2} \right)^3}{\left( {{t^2} + 1} \right)^2}\)
- \(\displaystyle f\left( w \right) = \frac{{{w^2} + 2w + 1}}{{3w - 5}}\)
- \(\displaystyle h\left( t \right) = \frac{{3 - 4t}}{{{t^2} + 1}}\)
- \(\displaystyle R\left( y \right) = \frac{{{y^2} - y}}{{{y^2} + 3y + 8}}\)
- \(Y\left( x \right) = \sqrt[3]{{x - 7}}\)
- \(f\left( t \right) = {\left( {{t^3} - 25t} \right)^{\frac{2}{3}}}\)
- \(h\left( x \right) = \sqrt[5]{x}\,\,{\left( {2x + 8} \right)^2}\)
- \(Q\left( w \right) = \left( {6 - {w^2}} \right)\,\,\,\sqrt[3]{{{w^2} - 4}}\)
- \(\displaystyle Q\left( t \right) = 7\sin \left( \frac{t}{4} \right) - 2\)
- \(g\left( x \right) = 3\cos \left( {2x} \right) - 5x\)
- \(f\left( x \right) = 7\cos \left( x \right) + 2x\)
- \(h\left( t \right) = 6\sin \left( {2t} \right) + 12t\)
- \(\displaystyle w\left( z \right) = {\cos ^3}\left( \frac{z}{5} \right)\)
- \(U\left( z \right) = \tan \left( z \right) - 4z\)
- \(h\left( x \right) = x\cos \left( x \right) - \sin \left( x \right)\)
- \(h\left( x \right) = 2\cos \left( x \right) - \cos \left( {2x} \right)\)
- \(f\left( w \right) = {\cos ^2}\left( w \right) - {\cos ^4}\left( w \right)\)
- \(F\left( w \right) = {{\bf{e}}^{14w + 3}}\)
- \(g\left( z \right) = {z^2}{{\bf{e}}^{1 - z}}\)
- \(A\left( x \right) = \left( {3 - 2x} \right){{\bf{e}}^{{x^{\,2}}}}\)
- \(P\left( t \right) = \left( {6t + 1} \right){{\bf{e}}^{8t - {t^{\,2}}}}\)
- \(f\left( x \right) = {{\bf{e}}^{3 + {x^{\,2}}}} - {{\bf{e}}^{2{x^{\,2}} - 4}}\)
- \(f\left( z \right) = {{\bf{e}}^{{z^{\,2}} - 4z}} + {{\bf{e}}^{8z - 2{z^2}}}\)
- \(h\left( y \right) = {{\bf{e}}^{6{y^{\,3}} - 8{y^{\,2}}}}\)
- \(g\left( t \right) = {{\bf{e}}^{2{t^{\,3}} + 4{t^{\,2}} - t}}\)
- \(Z\left( t \right) = \ln \left( {{t^2} + t + 3} \right)\)
- \(G\left( r \right) = r - \ln \left( {{r^2} + 1} \right)\)
- \(A\left( z \right) = 2 - 6z + \ln \left( {8z + 1} \right)\)
- \(f\left( x \right) = x - 4\ln \left( {{x^2} + x + 2} \right)\)
- \(g\left( x \right) = \ln \left( {4x + 2} \right) - \ln \left( {x + 4} \right)\)
- \(h\left( t \right) = \ln \left( {{t^2} - t + 1} \right) + \ln \left( {4 - t} \right)\)
- 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.
- 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.
- 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.