Let's start with information about where a function is increasing/decreasing/concave up/concave down, and see if we can use that to understand the behavior of the function. {{CMD_85}} Let {{MATH_86}} be a continuous function and suppose that: {{CMD_87}} {{CMD_88}} {{MATH_89}} for {{MATH_90}}. {{CMD_91}} {{MATH_92}} for {{MATH_93}} and {{MATH_94}}. {{CMD_95}} {{MATH_96}} for {{MATH_97}} and {{MATH_98}}. {{CMD_99}} {{MATH_100}} for {{MATH_101}}. {{CMD_102}} Sketch a possible graph of {{MATH_103}}. {{CMD_104}} Start by marking where the derivative changes sign and indicate intervals where {{MATH_105}} is increasing and intervals {{MATH_106}} is decreasing. The function {{MATH_107}} has a negative derivative from {{MATH_108}} to {{MATH_109}}. This means that {{MATH_110}} is {{CMD_111}}{{CMD_112}}increasing{{CMD_113}}{{CMD_114}}decreasing{{CMD_115}}{{CMD_116}} on this interval. The function {{MATH_117}} has a positive derivative from {{MATH_118}} to {{MATH_119}}. This means that {{MATH_120}} is {{CMD_121}}{{CMD_122}}increasing{{CMD_123}}{{CMD_124}}decreasing{{CMD_125}}{{CMD_126}} on this interval. Finally, The function {{MATH_127}} has a negative derivative from {{MATH_128}} to {{MATH_129}}. This means that {{MATH_130}} is {{CMD_131}}{{CMD_132}}increasing{{CMD_133}}{{CMD_134}}decreasing{{CMD_135}}{{CMD_136}} on this interval. {{CMD_137}} {{ENV_138}} {{CMD_139}} Now we should sketch the concavity: {{CMD_140}}{{CMD_141}}concave up{{CMD_142}}{{CMD_143}}concave down{{CMD_144}}{{CMD_145}} when the second derivative is positive, {{CMD_146}}{{CMD_147}}concave up{{CMD_148}}{{CMD_149}}concave down{{CMD_150}}{{CMD_151}} when the second derivative is negative. {{CMD_152}} {{ENV_153}} {{CMD_154}} Finally, we can sketch our curve: {{CMD_155}} {{ENV_156}} {{CMD_157}} {{CMD_158}} {{CMD_159}}