Since both local maximum and local minimum occur at a critical point, when we locate a critical point, we need to determine which, if either, actually occurs. {{CMD_319}} Find all local maximum and minimum points for the function {{MATH_320}}. {{CMD_321}} Write {{MATH_322}} We can easily express {{MATH_323}} as a product of its factors {{MATH_324}} which implies that the function {{MATH_325}} has only two critical points, {{MATH_326}} and {{MATH_327}}. Notice that the derivative {{MATH_328}} is a polynomial, and polynomials do not change signs except possibly at their zeros. This implies that derivative {{MATH_329}} {{CMD_330}}does not change the sign{{CMD_331}} on the intervals {{MATH_332}}, {{MATH_333}}, and {{MATH_334}}, because these intervals do not contain any zeros of {{MATH_335}}. {{CMD_336}} Select the correct statement about the sign of {{MATH_337}} on the intervals {{MATH_338}} and {{MATH_339}}. {{CMD_340}} {{CMD_341}}{{MATH_342}} on {{MATH_343}} and {{MATH_344}} on {{MATH_345}}. {{CMD_346}} {{CMD_347}}{{MATH_348}} on {{MATH_349}} and {{MATH_350}} on {{MATH_351}}. {{CMD_352}} {{CMD_353}}{{MATH_354}} on {{MATH_355}} and {{MATH_356}} on {{MATH_357}}. {{CMD_358}} {{CMD_359}}{{MATH_360}} on {{MATH_361}} and {{MATH_362}} on {{MATH_363}}{{CMD_364}} {{CMD_365}} {{CMD_366}} If we know the sign of the derivative on an interval, we also know whether the function is increasing or decreasing on that interval. This will help us determine whether the function has a local extremum at the critical point where {{MATH_367}}. {{CMD_368}} {{CMD_369}}